Thuener Armando da Silva

Optimization Under Uncertainty for AssetAllocation

TESE DE DOUTORADO

Thesis presented to the Programa de Pos–Graduacao emInformatica of the Departamento de Informatica da PUC-Rio as partial fulfillment of the requirements for the degreeof Doutor.

Advisor : Prof. Marcus Vinicius Soledade Poggi de AragaoCo–Advisor: Prof. Davi Michel Valladao

Rio de JaneiroApril 2015

Thuener Armando da Silva

Optimization Under Uncertainty for AssetAllocation

Thesis presented to the Programa de Pos–Graduacao emInformatica of the Departamento de Informatica of CentroTecnico Cientıfico da PUC-Rio, as partial fulfillment of therequirements for the degree of Doutor.

Prof. Marcus Vinicius Soledade Poggi de AragaoAdvisor

Departamento de Informatica – PUC-Rio

Prof. Davi Michel ValladaoCo–Advisor

Departamento de Engenharia Industrial – PUC-Rio

Prof. Helio Cortes Vieira LopesDepartamento de Informatica – PUC-Rio

Prof. Alexandre Street de AguiarDepartamento de Engenharia Eletrica – PUC-Rio

Prof. Vitor Luiz de MatosPLAN4

Prof. Geraldo Gil VeigaRN Tecnologia

Prof. Bruno da Costa FlachIBM

Prof. Jose Eugenio LealCoordinator of the Centro Tecnico Cientıfico da PUC-Rio

Rio de Janeiro, April 06, 2015

All rights reserved.

Thuener Armando da SilvaThuener Silva graduated as Bachelor in Computer Science atPUCRio in 2007. During graduate developed a project thatused machine learning techniques for sentiment analysis withRaul Renteria. In 2010, completed his Master in ComputerScience in Optimization and Automatic Reasoning area. De-veloped the thesis about the portfolio selection applied tothe Brazilian financial market entitled Experimental Studyof Techniques for Portfolio Optimization with his advisorEduardo Laber. During the Master received the scholarshipfrom CNPq and maintained an excellent academic perform-ance. His experience in Computer Science has an emphasis inAlgorithms, Machine Learning, Information Retrieval, Port-folio Selection and Quantitative Methods.

Bibliographic DataSilva, Thuener

Optimization Under Uncertainty for Asset Allocation/ Thuener Armando da Silva; Advisor: Marcus ViniciusSoledade Poggi de Aragao; Co–advisor: Davi MichelValladao. – 2015.

99 f: il. (color.) ; 30 cm

Tese (Doutorado em Informatica) – Pontifıcia Univer-sidade Catolica do Rio de Janeiro, Departamento de In-formatica, 2015.

Inclui bibliografia

1. Informatica – Teses. 2. Selecao de Carteiras. 3.Alocacao de Ativos em multi-estagio. 4. Analise de In-vestimentos. 5. Metodo de Apoio a Tomada de Decisao.6. Black Litterman. 7. Programacao Dinamica Dual Es-tocastica. I. Poggi, Marcus Vinicius. II. Valladao, DaviMichel. III. Pontifıcia Universidade Catolica do Rio deJaneiro. Departamento de Informatica. IV. Tıtulo.

CDD: 004

To my wife, my daughter and my family.

Acknowledgments

I would like first to thank my wife Luana, for your unconditional support

throughout this long journey. This work was only possible thanks to your help,

patience and understanding. During all these years by your side your kindness

and affection shown me the amazing person you are and how wonderful is our

love, you make my life better every day.

Family is the most important thing in my life, thanks to my family for

making me who I’ am, especially to my parents, Fernando and Katia, my

sisters and my brothers-in-law. Also, thank to my father-In-law, Gerson, for

the guidance and revision on this work.

Thanks to my advisor Marcus Poggi for the support over these years,

without your help this work would be impossible, you inspired me to be better

every day. Also, thanks to my co-supervisor David Valladao, for your patience

and dedication during this work, you believed in me even when I was very

skeptical, it is a great pleasure to work with you. I also want to thank the

professors Placido Pinheiro, Alexandre Street for their involvement in this

research.

Friends and colleagues make this difficult journey much easier, a special

thanks to my friends from Galgos laboratory and WhileTrue for the thoughts

and jokes.

Finally, I would like to thank all the teachers and staff at PUC-Rio and

the Department of Informatics for being part of this amazing institution that

supports my intellectual and personal growth and the PUC-Rio and CNPq for

the financial support.

In times we need the most is when we can clearly perceive that everything

we do comes back to us. I counted with the help of many people on things that

I thought would be impossible to do. I thank everyone who helped me in this

incredible journey.

AbstractSilva, Thuener; Poggi, Marcus Vinicius(Advisor); Valladao, Davi Michel.Optimization Under Uncertainty for Asset Allocation. Rio deJaneiro, 2015. 99p. DSc Thesis – Departamento de Informatica, PontifıciaUniversidade Catolica do Rio de Janeiro.

Asset allocation is one of the most important financial decisions made

by investors. However, human decisions are not fully rational, and people

make several systematic mistakes due to overconfidence, irrational loss aversion

and misuse of information, among others. In this thesis, we developed two

distinct methodologies to tackle this problem. The first approach has a more

qualitative view, trying to map the investor’s vision of the market. It tries to

mitigate irrationality in decision-making by making it easier for an investor to

demonstrate his/her preferences for specific assets. This first research uses the

Black-Litterman model to construct portfolios. Black and Litterman developed

a method for portfolio optimization as an improvement over the Markowitz

model. They suggested the construction of views to represent an investor’s

opinion about future stocks’ returns. However, constructing these views has

proven difficult, as it requires the investor to quantify several subjective

parameters. This work investigates a new way of creating these views by using

Verbal Decision Analysis. The second research focuses on quantitative methods

to solve the multistage asset allocation problem. More specifically, it modifies

the Stochastic Dynamic Dual Programming (SDDP) method to consider real

asset allocation models. Although SDDP is a consolidated solution technique

for large-scale problems, it is not suitable for asset allocation problems due

to the temporal dependence of returns. Indeed, SDDP assumes a stagewise

independence of the random process assuring a unique cost-to-go function

for each time stage. For the asset allocation problem, time dependency is

typically nonlinear and on the left-hand side, which makes traditional SDDP

inapplicable. This thesis proposes an SDDP variation to solve real asset

allocation problems for multiple periods, by modeling time dependence as a

Hidden Markov Model with concealed discrete states. Both approaches were

tested in real data and empirically analyzed. The contributions of this thesis

are the methodology to simplify portfolio construction and the methods to

solve real multistage stochastic asset allocation problems.

KeywordsPortfolio Selection; Multistage Asset Allocation; Investments Analysis;

Decision support systems; Black Litterman; Stochastic Dual Dynamic Pro-

gramming.

Resumo

Silva, Thuener; Poggi, Marcus Vinicius; Valladao, Davi Michel. Otimi-zacao Sob Incerteza para Alocacao de Ativos. Rio de Janeiro,2015. 99p. Tese de Doutorado – Departamento de Informatica, PontifıciaUniversidade Catolica do Rio de Janeiro.

A alocacao de ativos e uma das mais importantes decisoes financeiras

para investidores. No entanto, as decisoes humanas nao sao totalmente racion-

ais. Sabemos que as pessoas cometem muitos erros sistematicos como, excesso

de confianca, aversao a perda irracional e mau uso da informacao entre outros.

Nesta tese desenvolvemos duas metodologias distintas para enfrentar esse prob-

lema. A primeira abordagem e qualitativa, utiliza o modelo de Black-Litterman

e tenta mapear a visao que o investidor tem do mercado. Esse metodo tenta

mitigar a irracionalidade na tomada de decisao tornando mais facil para um in-

vestidor demonstrar suas preferencias em relacao aos ativos. Black e Litterman

desenvolveram um metodo para otimizacao de carteiras com a proposta de mel-

horar o modelo Markowitz, utilizando a construcao de visoes para representar

a opiniao do investidor sobre o futuro. No entanto, a forma de construir essas

visoes e bastante confusa e exige que o investidor estime varios parametros

que sao subjetivos. Assim, propomos uma nova forma de criar essas visoes,

utilizando Analise Verbal de Decisao. A segunda pesquisa envolve metodos

quantitativos para resolver o problema de alocacao de ativos com multiplos

estagios com premissas mais realistas. Embora a Programacao Dinamica Dual

Estocastica (PDDE) seja uma tecnica promissora para a solucao de problemas

de grande porte, nao e adequada para o problema de alocacao de ativos devido

a dependencia temporal associada aos retornos dos ativos. PDDE assume que

o processo estocastico tem independencia por estagio assegurando uma funcao

unica de custo futuro para cada estagio. No problema de alocacao de ativos, a

dependencia do tempo e tipicamente nao-linear e no lado esquerdo, o que torna

PDDE tradicional nao aplicavel. Propomos uma variacao do PDDE usando

modelo oculto de Markov com estados discretos para resolver problemas reais

de alocacao de ativos com multiplos perıodos e dependencia no tempo. Ambas

as abordagens foram testadas em dados reais e empiricamente analisadas. As

principais contribuicoes sao as metodologia desenvolvidas para simplificar a

construcao de portfolios e para resolver o problema de alocacao de ativos com

multiplos estagios.

Palavras–chaveSelecao de Carteiras; Alocacao de Ativos em multi-estagio; Analise

de Investimentos; Metodo de Apoio a Tomada de Decisao; Black Litterman;

Programacao Dinamica Dual Estocastica.

Contents

1 Introduction 13

1.1 Objective 13

1.2 Contributions 14

1.3 Outline 15

1.4 Assumptions and Notation 15

2 Asset Allocation 17

2.1 Utility function 17

2.2 Mean-Variance Model 18

2.3 Modern Asset Allocation Methods 19

3 More Human-like Portfolio Optimization Approach 21

3.1 Introduction 21

3.2 Verbal Decision Analysis 23Formal Statement of the Problem 25The ZAPROS-III Method 25

3.3 Black-Litterman 26Market Equilibrium 27Specifying Views 29The Estimation Model 30Idzorek 31

3.4 Experiments with Brazilian stocks 31Construction of the Views 31Results 32

4 Dynamic Asset Allocation Under Uncertainty 39

4.1 Introduction 39

4.2 Stochastic Dynamic Dual Programming 41Risk Neutral SDDP 42Time Consistent Risk Averse Model 43

Conditional value at risk 44Stopping Criteria 46Sampling Scenarios 47

4.3 Stochastic Dynamic Programming for Asset Allocation 49Myopic policy: No transaction costs and temporal independence 50SDDP for asset allocation: Transaction costs and temporal independence 51

Transactional costs 52H2SDDP for allocation: Transaction costs and temporal dependence 53

H2SDDP 54Robust H2SDDP for asset allocation: Transaction costs, temporal depend-ence and ambiguity aversion 61

H2SDDP for asset allocation: Transaction costs, temporal dependence andsell short 63

4.4 Computational Experiments 64Data analysis 64Sampling Methods 67Sensitivity Analysis 68

HMM 69Impact of the Risk Aversion 69Convergence and trials 70Transactional costs 71

Models Evaluation 73Experiment with Monthly Data Set 2012 to 2014 74Experiment with Monthly Data 2007 to 2014 76Experiment with Daily Data Set 80

5 Conclusions and Future Works 82

Bibliography 86

A Appendix 95

A.1 Questionnaires 95

A.2 Myopic prove 96

List of Figures

3.1 Procedure to apply ZAPROS-III methodology 263.2 Flowchart of Black-Litterman method [19] 273.3 The Black-Litterman portfolio with our views 353.4 Equilibrium portfolio 353.5 Result of the increase in the qualification of Sid. Nacional 363.6 Result of the increase in the qualification of Oi 363.7 Result of right scenario, where the investor guess is right 383.8 Result of wrong scenario, where the investor guess is wrong 38

4.1 -CVaR and -VaR for gain distribution 454.2 Example of Latin Hypercube Sampling for uniform distribu-

tion with two dimensions 484.3 Sampling with Monte Carlo 494.4 Sampling with Latin Hypercube Sampling 494.5 LHS interval for a normal distribution N ∼ (0, 1) 494.6 An example of HMM for stock market 564.7 An example of a Gaussian mixture model. 564.8 Decision tree of the generic problem with return dependence 604.9 Decision tree of the problem with return dependence

modeled with HMM 604.10 Decision tree of the problem of our proposal 614.11 Metrics for historical monthly return series of the five indus-

trial portfolios 654.12 Metrics for historical daily return series of the ten industrial

portfolios 664.13 Cumulative performance for monthly data set of the five

industrial portfolios 664.14 Cumulative performance for daily data set of the ten indus-

trial portfolios 674.15 Comparison between Latin Hypercube Sampling (LHS) and

Monte Carlo (MC) 684.16 States likelihood for train data 694.17 Impact of the risk aversion coefficient (λ) on the upper bound

for monthly returns and daily returns, respectively on the leftand right 70

4.18 Convergence of the Upper Bound for monthly data set 714.19 Impact of the transactional costs on the upper bound 724.20 Portfolio allocation without transactional costs 734.21 Portfolio allocation with transactional costs 734.22 Comparing the different methods for asset allocation data

form 2012 to 2014 754.23 Cumulative returns series from January 2012 to December

2014 754.24 Returns series from January 2007 to December 2014 76

4.25 Comparing the different methods for asset allocation dataform 2007 to 2014 77

4.26 Comparing the market states with the performance of theH2SDDP for data form 2007 to 2014 78

4.27 Trailing returns for asset allocation methods with monthlydata form 2007 to 2014 78

4.28 Comparing the performance of H2SDDP with sell short sellshort with other methods, monthly data form 2007 to 2014 79

4.29 Trailing returns of H2SDDP with sell short sell and othermethods, monthly data form 2007 to 2014 80

4.30 Cumulative performance for asset allocation methods withdaily data set form 2007 to 2014 81

4.31 Trailing returns for asset allocation methods with daily dataset form 2007 to 2014 81

List of Tables

1.1 Set and stochastic process notation 151.2 State and decision variable notation 16

2.1 Some properties of the utility function 18

3.1 FIQ and sector of the stocks 333.2 FIQ of the sectors 333.3 The expected return of the stocks 343.4 A summary of the views data 343.5 stocks returns for the period 373.6 Sectors returns for the period 373.7 Return for the different scenarios and the Market Portfolio 38

4.1 Monthly data series for 5 industrial portfolios 654.2 Daily data series for 10 industrial portfolios 654.3 Number of stages and the computational time 704.4 CVaR and return values of the simulated polices for different

risk coefficients, results in percentage 74

A.1 Questionnaire about the stocks 95A.2 Questionnaire about the sectors 96

1Introduction

Asset allocation is one of the most important financial decisions made

by investors. The asset allocation problem consist in finding a portfolio (for

example stocks, bonds, cash and gold) that better suits the investor’s needs.

Selecting good portfolios represents a competitive advantage, and to make

good decisions in this regard it is necessary to leave emotions aside.

Emotions often affect investment decisions. In most cases, human de-

cisions are not fully rational and people often make systematic mistakes due

to overconfidence, irrational loss aversion and misuse of information. Such mis-

takes, made frequently by investors, can lead to big financial losses, which is

one of the reasons why the behavioral finance field is dedicated to analyzing the

psychology of financial decision making. Hence the need for tools that support

the investors’ financial decisions and prevent pitfalls.

Investment analysis techniques can be used as a tool or as automated

optimization models to minimize, as much as possible, irrational intervention

in decision making. In the last decades, there has been a remarkable increase

in the use of financial models and optimization techniques for asset allocation.

One of the main reasons for this is the attractive assumption that it is possible

to forecast the conditional moments of the return distributions [1]. Another

reason is the growth in processing power and the development of methods and

optimization solutions that can handle a large volume of data. The field hardly

existed in 1980, but has experienced a rapid surge ever since. Every day more

tools are used to support the creation of investment strategies, and currently

there is a large variety of approaches to the problem; Robust Optimization,

Stochastic Programming and Machine Learning being only a few of the many

fields in which we can encounter solutions to assist in financial decision making.

1.1 ObjectiveThe main objective of this thesis is to help investors solve real asset

allocation problems. In this regard, it presents two alternatives that aim at

helping financial decision making.

The first approach has a qualitative perspective, trying to map the

investor’s vision of the market. It is an attempt to mitigate irrationality

Chapter 1. Introduction 14

in decision making, making it easier for investors to demonstrate reasoned

preferences concerning assets. For this purpose, the method combines the

Black-Litterman portfolio optimization method with verbal decision analyses.

However, even when using decision-support methods, we are not safe

from irrational decisions. That is why this work proposes quantitative methods

using stochastic models to evaluate and solve the multistage asset allocation

problem. More specifically, it suggests modifying the Stochastic Dynamic Dual

Programming method to consider real asset allocation models.

1.2 ContributionsThe major contributions of this work are derived from the two proposed

methodologies. The first part of this thesis contributes by developing a simple

new methodology that fits the investor’s needs based on verbal decision

analyses and Black-Litterman. This work has shown that it is possible to

optimize portfolios even when the investor is not an expert on the subject. In

addition, this approach makes it easier for the investors to manifest their own

opinion, in an organized fashion, and allows them to change their portfolios

more frequently. Finally, a case study based on Brazilian stocks demonstrates

that this methodology creates more intuitive and diversified portfolios.

The second part of this thesis proposes a new approach to solve

multistage stochastic asset allocation problems with time dependency. The

method maps the temporal dependence as hidden Markovian states, trans-

forming them into a convex problem that can be solved by adapting the SDDP.

In addition, it presents a more general model that consider the ambiguity

on the states’ probabilities of the optimization problem. As our experiments

demonstrate, the proposed model performs very well and shows promising

results.

Main contributions:

– Proposes a new way to generate the Black-Litterman views using VDA,

enabling the investor to construct personalized portfolio based on the

his/her opinion.

– Model the multistage stochastic portfolio optimization problem with

hidden Markovian temporal dependence and transactional cost.

– Create a more general model for the multistage stochastic portfolio op-

timization problem with ambiguity aversion, with the intent to mitigate

returns estimation errors.

Chapter 1. Introduction 15

1.3 OutlineThis work is organized as follows: Chapter 2 introduces asset allocation

and gives an overview of the methods that have been used to approach this

problem. Chapter 3 is dedicated to the first proposed methodology, providing

an overview of the Verbal Decision Analysis and Black-Litterman, and of how

these two methods are combined. It also presents experiments conducted in the

Brazilian stock market and some remarks about the proposed methodology.

Chapter 4 describes the second, more quantitative, methodology proposed

in this work. Its first part introduces the concept of SDDP and explains

how this method may be adapted for asset allocation, while its final part

proposes alternative models and shows various computational experiments.

Chapter 5 brings this thesis’ final conclusion and debate regarding future

works, presenting the main contributions of the two proposed methodologies

and suggestions for future works.

1.4 Assumptions and NotationIn this section, we present some assumptions and notation used through

this thesis. It will be used bold-faced upper (Σ, Π, Q, P, . . .) and lowercase (µ,

p, r, . . .) letters to denote, respectively matrices and vectors [2]. To simplify

the formulations it will be use a vector with all elements equal one with proper

dimension 1 = [1, . . . , 1]>.

The multistage problem has a finite planning horizon T ; the probability

space is (Ω,F ,P) with filtration F , where F = ∅,Ω and F = FT . A specific

notation for the portfolio selection application was created and is shown in

Table 1.1 and Table 1.2.

Sets

A = 1, . . . , A: Index set of the A ≥ 1 assets.

H = 0, . . . , T − 1: Set of stages.

Stochastic Process

rti (s): Excess return of asset i ∈ A, between stages t ∈1, . . . , T and t − 1, under scenario s ∈ Ω, where

rt (s) = (r1,t (s) , . . . , rA,t (s))> and r[t′,t] (s) =

(rt′ (s) , . . . , rt (s))> for t′ ≤ t .

r[t′,t] = (rt′ , . . . , rt)>: Realization sequence of the asset returns for t′ ≤ t.

r[t] = r[0,t]: Realization sequence of the asset returns for 0 to t.

Table 1.1: Set and stochastic process notation

Chapter 1. Introduction 16

State Variables

Wt (s): Wealth at stage t ∈ H ∪ T under scenario s ∈ Ω

Decision Variables

xti (s): Amount invested in asset i ∈ A, at stage t ∈ H under scenario

s ∈ Ω, where xt (s) = (x1,t (s) , . . . , xA,t (s))> and x[t′,t] (s) =

(xt′ (s) , . . . ,xt (s))> for t′ ≤ t

Table 1.2: State and decision variable notation

Without loss of generality, the risk-free asset is represented as the first

asset of the portfolio without excess return, i.e., for each scenario r1,t (s) = 0,

for all t ∈ H ∪ T and all s ∈ Ω.

2Asset Allocation

Before Markowitz’s work [3] the concept of diversification was greatly

simplified. People had an intuition that putting everything in one asset lead

to largely different results, while distributing their money among many assets

greatly reduced the chance of all having bad returns simultaneously.

The essence of portfolio optimization theory is based on diversification,

as by combining different assets it is possible to obtain a lower risk than the

offered by any of the assets individually. As the number of the assets increase,

the variance of the portfolio decreases towards zero [1].

The Asset Allocation consists in finding the most appropriate group of

assets while considering the individual properties of each asset. The optimal

portfolio varies according to the profile of each investor, and there isn’t a single

portfolio that is recommended for every type of investor. This is due to the

specific characteristics of each individual, institution or group. An investor

averse to risk may prefer to invest in assets with low risk and low return, while

another investor, who is more open to risk, might prefer assets with more risk

when it is possible to achieve higher returns.

2.1 Utility functionTo provide the most suitable portfolio for a given investor, it is necessary

to understand its preference among the set of possible investments. A way

to accomplish this is to map the utility function U that captures investor’s

satisfaction or happiness at a given level of wealth. For a given level of

wealth W , the usefulness U(W ) is the investor’s satisfaction achieved with

this wealth. Assuming that an investor with utility function U attempts to

make an investment portfolio and that P (ws) is the probability of this portfolio

generating a wealth ws for a scenario s, we can calculate the expected utility

for the portfolio as being:

E[C] =∑s

U(ws)P (ws)

Utility functions have some important properties. First, the utility func-

tion should prefer more to less wealth, and therefore the utility of X + 1 units

Chapter 2. Asset Allocation 18

is greater than that of X units. This implies a positive first derivative of the

utility function. The investor’s attitude towards risk is also related to the util-

ity function. Considering that when it comes to risk, an investor can be averse

to it, neutral or like it.

The investor’s classification can be defined by considering a fair bet, i.e.

an investment where the value is equal to the expected cost. The investor

that rejects a fair bet is risk averse, and thus its dissatisfaction with a loss is

greater than its satisfaction with a gain for the same value. Functions with this

behavior have a negative second derivative. The opposite occurs in the case

of an investor who accepts a fair bet, a case in which the second derivative

is positive. For an investor who is indifferent to risk, the utility of these two

investments, with the same expected value, is the same, meaning the second

derivative of this function is zero. Table 2.1 presents the characteristics of the

second derivative according to investors’ profiles.

Profile Behavior Implication

Risk aversion Rejects fair bet U′′(W ) < 0

Risk neutral Indifferent to fair bet U′′(W ) = 0

Taste for Risk accepted the fair bet U′′(W ) > 0

Table 2.1: Some properties of the utility function

2.2 Mean-Variance ModelIn his pioneering work “Portfolio Selection” [3], Markowitz developed a

model that allows the selection of portfolios considering the relation between

return and risk. This became known as the mean-variance model, which uses

the expected return as a measure of performance of the portfolio and the

variance as a risk measure.

A portfolio with n assets can be represented by the amount invested

on each asset x = (x1, . . . , xA). Assuming that future returns are random

variables and using each individual expected return, the portfolio return can

be estimated by the equation

E[Rp] =A∑i=1

xiE[Ri]

In order to evaluate the variance of portfolio σ2c , it is used the covariance

of the return series between all pairs of assets that make up the portfolio

σij, ∀i, j ∈ 1, . . . , A.

Chapter 2. Asset Allocation 19

σ2p =

A∑i=1

A∑j=1

(xixjσij)

Using the mean-variance approach, one can create an efficient frontier.

That is the set of portfolios in which a given risk level has the highest possible

return, or a certain level of return has the lowest possible risk. These are

efficient portfolios. All other portfolios can be considered inefficient, because

they have either lower return or higher risk.

Formally, the two minimization (2.1) and maximization (2.2) formula-

tions are described below

minx

A∑i=1

x2iσ

2i +

A∑i=1

A∑j=1,j 6=i

xixjσij (2.1)

s. t.A∑i=1

µixi ≥ rp

A∑i=1

xi = 1

xi ≥ 0, ∀i

maxx

A∑i=1

µixi (2.2)

s. t.A∑i=1

x2iσ

2i +

A∑i=1

A∑j=1,j 6=i

xixjσij ≤ vp

A∑i=1

xi = 1

xi ≥ 0, ∀i

where the number of assets that may be part of the portfolio is N , and xi is

the percentage of the portfolio that will be granted to asset i. The covariance

between assets i and j is represented by σij, and σ2i is the variance of asset i.

Finally, µi is the expected return of asset i, and rp and vp are the minimum

return and the maximum risk for the desired portfolio, respectively.

2.3 Modern Asset Allocation MethodsEven with several tools available to help investors create mean-variance

optimized portfolios, they are still very skeptical about the mean-variance

theory and its practical implications. Albeit revolutionary, Markowitz’s work

Chapter 2. Asset Allocation 20

has shown major drawbacks in practical applications, yielding portfolios that

can be counter-intuitive [4, 5], that tend to concentrate on a small subset

of available assets and that do not seem well diversified [6, 7]. The optimal

portfolio is also extremely sensitive to small variations in the input data [5,6,8].

This does not mean, however, that the mean-variance theory is flawed,

but only that the idea needs to be remodeled or adapted in order to achieve

better results. Hence, several new methodologies for portfolio optimization,

and consequently for asset allocation, have been developed.

For instance: regarding the mean-variance’s hypersensitivity to changes

on the estimated inputs, which suggests that those parameters need to be

estimated in an extremely precise way, several attempts to reduce the impact

of estimation errors have been made. In fact, there are several ways to

create portfolios that are less sensitive to these variations, such as shrinkage

estimators, Bayesian and resampling methods and robust optimization [4, 5,

9, 10]. Frost and Savarino [11] have demonstrated that these optimization

constraints stabilize the portfolio and generally improve performance. As

mentioned by Jagannathan and Ma [12], these constrains can be interpreted

as a posteriori regularization.

In this thesis, we will focus on two different approaches, presented in

the following chapters. Considering the instability problems and parameter-

estimation method of the Markowitz model, we will propose a methodology to

construct a portfolio based on the investors’ opinion. Another approach is to

focus on solving the asset allocation problem in a quantitative way. This will

be modeled as a multistage stochastic problem with temporal dependence.

3More Human-like Portfolio OptimizationApproach

3.1 IntroductionThe practical disadvantages of the Markowitz model motivated Fisher

Black and Robert Litterman to develop a new approach. Thus the Black-

Litterman approach [13], which combines the expected equilibrium between

returns estimated through the Capital Asset Pricing Model (CAPM) and views

to optimize the portfolio. The views represent the investor’s opinion about the

stocks’ future returns. This model yields more stable and diversified portfolios

than the mean-variance standard model [14].

Black and Litterman’s original paper [4] only explained the core aspects

of their idea, leaving it to others to better explain the implication of their

model. Satchell and Scowcroft [15], Walters [14], He and Litterman [16] explain

the Black-Litterman solution in further detail. Walters [14] also constructed

a framework1 to use the model and other portfolio optimization techniques.

Mankert [17] sheds more light on the practical implications of the Black-

Litterman approach. Other studies focus on extensions of the original model,

like Herold [18], Idzorek [19], Fernandes et al. [20], and Meucci [21].

Also, Bertisimas et al. [2] proposed a more general extension of the ori-

ginal Black-Litterman model that can incorporate investor opinion about volat-

ility and construct estimators for more general notions of risk. Reinterpreting

the problem through inverse optimization Bertisimas et al. [2] extends the

traditional model creating a approach that can combine a greater variety of

views.

The expression of the investor’s preferences can be seen as a decision

making process. Traditionally, decision making scenarios involve the analysis

of objects from several points of view and can be assisted by multi-criteria

methodologies. These help generating knowledge about the decision context

and, as a consequence, increase the confidence of those making decisions [22].

There are multi-criteria methods based either on quantitative or qualitative

1That is available in www.blacklitterman.org

Chapter 3. More Human-like Portfolio Optimization Approach 22

analysis of the problem, and choosing the best approach is a great challenge.

Examples of problem-solving using quantitative methods can be found in

Castro et al. [23], Toncovich et al. [24], and Pinheiro et al. [25]. Among those

who apply qualitative methods, we have Mendes et al. [26], Tamanini et al. [27],

Tamanini et al. [28], Tamanini et al. [29] and Castro et al. [30].

The Verbal Decision Analysis is based on multi-criteria problem-solving

through qualitative analysis methods. One of the advantages of qualitative

methods is that all the questioning in the process of eliciting preferences is

made in the decision maker’s native language. Moreover, verbal descriptions

are used to measure preference levels. This procedure is psychologically valid,

respecting the limitations of the human information processing system. This

characteristic makes the incomparability cases [31] become almost unavoidable

since the scale of preferences is purely verbal and consequently not an accurate

way of estimating values. Therefore, the method may not be capable of

achieving satisfactory results in some situations, presenting an incomplete

solution to the problem.

Establishing views in the traditional quantitative way is not an easy task

and an investor would need help from an expert in the process. That is why we

chose a method to setting views using Verbal Decision Analysis (VDA). For this

propose, we developed questionnaires that are intuitive and can be answered

by anyone with basic knowledge of investment options without needing any

further special training.

The purpose of this chapter is to develop a methodology that constructs

a personalized portfolio based on the investor’s opinions. Our problem is not a

typical multi-criteria problem, being actually very different from normal VDA

applications. This is one of the major difficulties that have to be overcome in

order to create the Black-Litterman views. For this purpose, in the final part of

Section 3.4 we compare the return of investing on the investor most preferred

asset with our proposed approach.

Moreover, the objective pursued is a technique to support the creation

of a personal portfolio based on an individual’s opinion, preferences or view.

Therefore, a comparison among performances of portfolios, in the present case,

should only consider portfolios that are aligned with the considered individual

preference. The technique proposed here follows the mean-variance balance of

Markowitz generated portfolios.

In Section 3.2 we present a brief explanation of the Verbal Decision

Analysis (VDA) framework used in this work. Section 3.3 brings a review

of the Black-Litterman methodology. Finally, in Section 3.4 we report about

the experiments made with Brazilian stocks.

Chapter 3. More Human-like Portfolio Optimization Approach 23

3.2 Verbal Decision AnalysisA decision may be defined as the result of a process of choice when

someone is confronted with a problem or with an opportunity for creation,

optimization or improvement of a given situation. On the other hand, decision

making is a special activity of human behavior, aimed at the achievement of

a given goal. It takes place in every activity of the human world, from simple

daily problems to complex situations inside an organization. The conclusion of

a decision making process can be an ordination of alternatives or the selection

of a single alternative from a list of possible solutions for the problem.

Establishing its preferences and interests is usually enough to allow an

individual to make decisions that solve simple problems. However, individuals

often find it hard to separate emotions from reason. As a result, emotions

often influence the decision making process [32,33]. The decision also involves

several factors, some of which may not be measurable. Thus, when a decision

maker needs to solve complex problems, covering many alternatives and a large

volume of information that may not be measurable nor easily comparable, some

methodologies exist to support the decision making process.

In order to solve a given problem, alternative solutions are taken into

consideration. Such alternatives are defined and characterized according to a

set of criteria, structured around its verbal and qualitative nature. There are

a huge number of practical problems which is necessary to generate an ordinal

scale of alternatives [34]. The construction of such an ordinal scale is helpful

in many situations, for example, to reject less preferable alternatives from a

given set.

The Verbal Decision Analysis (VDA) framework is a set of methods

defined to support the decision making process through the verbal representa-

tion of problems. Some methods that constitute the Verbal Decision Analysis

framework are: ZAPROS-III, ZAPROS-LM, PACOM, and ORCLASS Larichev

and Moshkovich [34]. According to Gomes et al. [35], in the majority of multi-

criteria problems there is a set of alternatives that can be evaluated against the

same set of characteristics (called criteria or attributes). The VDA framework

is structured on the supposition that most decision making processes can be

qualitatively described [36]. Although the decision maker’s ability to choose is

very dependent on the occasion and the stakeholders’ interest, the methods to

support decision making are universal.

Moreover, in Ustinovich and Kochin [37] the analysis of a large amount of

data-processing performed by human beings has shown that the psychologically

correct operations are:

Chapter 3. More Human-like Portfolio Optimization Approach 24

– Comparison of two assessments in verbal scale by two criteria;

– Assignment of multi-criteria alternatives to decision classes;

– Comparative verbal assessment of alternatives according to separate

criteria.

This last operation is the only classification methodology within the VDA

framework. The goal of the Verbal Decision Analysis framework is to establish

a ranking of alternatives in order of preference.

The methods belonging to the Verbal Decision Analysis framework may

be evaluated in light of their objectives:

– As a tool for ordinary classification, ORCLASS was one of the first

methods designed to tackle classification problems. There are several

other widely known methods for solving classification problems that can

be applied and analyzed for future applications [38–40], but that does

not belong to Ustinovich and Kochin’s [37] VDA framework;

– The other objective is to organize the solutions alternatives for the

problem in a rank, from the most preferable to the least preferable one.

Three methods are proposed within the VDA framework: ZAPROS-LM,

ZAPROS-III, and PACOM. Although they have the same final goal, they

have different purposes:

– PACOM is exclusively created to be applied according to pair com-

pensation and consists in comparing the advantages and disadvant-

ages of multi-attribute alternatives.

– The ZAPROS method was created to be applied by pair comparison

and consists in comparing a pair of alternatives with the advantage

of reaching a decision by using simple and understandable dialogue.

It is also divided in two alternative methods:

∗ ZAPROS-III differs from ZAPROS-LM in its level of treatment

of inconsistence. ZAPROS-III can be considered an evolution

of ZAPROS-LM in this concept.

Chapter 3. More Human-like Portfolio Optimization Approach 25

(a) Formal Statement of the Problem

The methodology follows the same problem formulation proposed by [34],

where:

1. K = c1, c2, . . . , cN, representing a set of N criteria;

2. nq represents the number of possible values on the scale of q-th criterion,

(q ∈ K); For the ill-structured problems, as in this case, usually nq ≤ 4;

3. Xq = x1, x2, . . . , xnq represents a set of values to the q-th criterion,

which is this criterion scale; |Xq| = nq; The values of the scale are ranked

from best to worst, and this order does not depend on the values of other

scales;

4. Y = X1×X2×· · ·×XN represents a set of vectors yi, in such a way that:

yi = yi1, yi2, . . . , yiN> and yi ∈ Y , yiq ∈ Xq, where |Y| =∏N

q=1 nq;

5. Z = zij1 and zi ∈ Y , where the set of j vectors represents the

description of the real alternatives.

The order of the multi-criteria alternatives on set A is defined based on the

decision maker’s preferences.

(b) The ZAPROS-III Method

According to Ustinovich and Kochin [37], one of the most important

features of ZAPROS methods is the use of psychologically grounded procedures

for identifying the preferences. This method evaluates personal abilities and

limitations of human information processing system. The disadvantages of the

method also include the limited amount of attributes and difficulties in using

quantitative criteria.

Furthermore, ZAPROS-III [33] considers values known as Quality Vari-

ations (QV) or Quality Changing (QC) [36] and Formal Index of Quality (FIQ).

The QV represents the distance between the evaluations of two criteria. The

FIQ mainly aims at minimizing the number of comparable pairs of alternatives.

The FIQ is used in the ranking of the alternatives.

Figure 3.1 [31] presents a flowchart with steps for the application of

the VDA method ZAPROS-III. As described in the Figure 3.1, the method’s

application can be divided into four stages: Problem Formulation, Elicitation

of Preferences/Comparison of Alternatives, Validation of the Decision maker’s

preferences, and Comparison of Alternatives.

Chapter 3. More Human-like Portfolio Optimization Approach 26

Figure 3.1: Procedure to apply ZAPROS-III methodology

A disadvantage of the method is that the number and values of criteria

that can be handled are limited, in order to keep complexity under control.

Tamanini [31] defends that although ZAPROS-III-i follows a procedure

similar to its predecessor’s to extract preferences, it also implements modifica-

tions that make it more efficient and more accurate regarding inconsistencies.

The number of incomparable alternatives is essentially smaller than in previous

ZAPROS [36].

3.3 Black-LittermanThe traditional portfolio approach proposed by Markowitz has some

issues and does not consider the investor’s vision of the market. Hence, the

Black-Litterman [13] was conceived to be a more practical and more flexible

portfolio management method [41]. Its methodology begins by determining the

equilibrium portfolio and the views of the investor, after these are combined to

construct a new distribution of the stocks’ returns. Using this new distribution,

a portfolio optimization problem is formulated and a new optimal portfolio is

obtained. A summary of the Black-Litterman model is present in Figure 3.2.

Chapter 3. More Human-like Portfolio Optimization Approach 27

Covariance

Matrix (Σ)

Market

capitalization

Weights (xm)

Risk Aversion

Coefficient

(δ)

Implied Equilib-

rium Return Vector

Π = δΣxm

Prior Equilibrium Distribution

N ∼ (Π, τΣ)

Views (Q)

Uncertainty

of Views

(Ω)

View Distribution

N ∼ (Q,Ω)

New Expected Return

Distribution N ∼ (µ,M)

New Return Distri-

bution N ∼ (µ, Σ)

Figure 3.2: Flowchart of Black-Litterman method [19]

The model proposed by Black and Litterman can be seen, in a rather

simplistic way, as an adjustment in the prior distribution of the assets’ returns

to adapt it to the investor’s vision. Essentially, however, it combines the

investor’s views with the CAPM notion of market equilibrium [4,13].

(a) Market Equilibrium

The Black-Litterman assumption is that the a priori distributions of

returns are consistent with market equilibrium. Considering that all investors’

utility functions are the same, the CAPM theory shows that everyone should

hold the same portfolio, the market portfolio xm. The market portfolio is the

portfolio where the amount of assets is proportional to its market value.

First we have to assume that the returns of the stocks r are normally

distributed with mean E(r) and covariance matrix Σ i.e. r ∼ N(E(r),Σ).

When the market is efficient, the expected return for any asset has the following

propertyE(ri)− rf = βi(E(rm)− rf ) (3.1)

Chapter 3. More Human-like Portfolio Optimization Approach 28

The E(ri) and E(rm) are the asset i and market portfolio’s expected returns,

while rf is the risk free asset return. Coefficient βi is the covariance between

asset i and the market portfolio returns, divided by the market portfolio

varianceβi =

σimσ2m

(3.2)

Also, the market portfolio return is

rm =n∑j=1

rjxmj (3.3)

The risk equilibrium premium Π is the expected excess of return yielded

by the risky stocks, which should perform better than the risk free stock. It

is properly defined as the difference between the asset returns and risk free

returns Πi = E(ri)− rf . Using the fact that

σim =n∑j=1

xmjσji (3.4)

and (3.1) we have

Πi = βi(E[rm]− rf ) (3.5)

=σimσ2m

(E[rm]− rf )

=E[rm]− rf

σ2m

(n∑j=1

xmjσji)

With the risk aversion parameter δ

δ =E[rm]− rf

σ2m

(3.6)

the final result can be expressed in matrix form as

Π = δΣxm (3.7)

A more detailed demonstration of these equations and more about the

CAPM theory can be found in [42] and [43]. The result above can also be

obtained by deriving the traditional quadratic utility function of the mean-

variance model, assuming that all investors solve this problem.

Finally, we can define the prior distribution as the real µ return distri-

bution with mean Π and variance τΣ

µ = Π + επ

επ ∼ N(0, τΣ) (3.8)

Chapter 3. More Human-like Portfolio Optimization Approach 29

The τ is a small number that reflects the investor’s uncertainty about prior

return estimations [44]. It is the most confusing parameter of the model and

has several different calibration approaches. Further ahead we shall present

Idzorek’s technique to eliminate τ .

(b) Specifying Views

The views are the investor’s vision regarding future market behavior.

These views can be relative or absolute and need to be “fully invested”. Hence,

the sum of weights is zero for the relative view, and one for the absolute. An

example of absolute view is “Stock i will return q1%” and of a relative view is

“International stock will outperform domestic stock by q2%”. Furthermore, the

confidence has to be defined by the investor, and this will change how much

the view will affect the portfolio weights. The investor’s view can be expressed

asPµ = Q + εq (3.9)

Where P is the perspective of the investor and Q specifies the expected

return of each view. The εq is an non-observable random and normally

distributed vector with mean zero and a diagonal covariance matrix Ω that

expresses the uncertainty of the views (εq ∼ N(0,Ω)).

Considering v as the number of views and n the number of stocks, P

will be a matrix v × n with pi (i ∈ 1, . . . , v) representing a vector with n

elements, Q a vector with v elements and Ω a v × v diagonal matrix

PT = [p1,p2,p3, . . . ,pv]

QT = [q1, q2, q3, . . . , qv]

Ω =

ω1 0 . . . 0

0 ω2 . . . 0...

.... . .

...

0 0 . . . ωv

To better understand how to describe these views in matrix form, two

views were created: one relative and other absolute. In the first view, stock one

will outperform stock two by 1% and in the second view, stock three will have

return 2%. [1 −1 0

0 0 1

]µ =

[0.01

0.02

]+ εq

Chapter 3. More Human-like Portfolio Optimization Approach 30

(c) The Estimation Model

With the expected excess return and the views of the investor, it is

possible to proceed to the next step of the Black-Litterman approach, which

combines these two items. There are two ways to estimate the final model. The

original Black-Litterman paper [13] references the Theil’s Mixed Estimation

model [45], but there is also a Bayesian approach. The first method was

chosen because it is easy to understand. By applying the identity matrix I,

the problem can be seen in the matrix form[I

P

]µ =

[Π

Q

]+

[επ

εq

](3.10)

Constructing the auxiliary matrices D =

[I

P

], C =

[Π

Q

]and ε =

[επ

εq

]we can reformulate the problem as

Dµ = C + ε (3.11)

ε ∼ N(0,W), W =

[τΣ 0

0 Ω

](3.12)

Solving this system of equations using least squares, we have

µ = (DTW−1D)−1DTW−1C

= [(τΣ)−1 + PTΩ−1P]−1[(τΣ)−1 + PTΩ−1Q] (3.13)

The variance can also be adjusted to reflect the change in the return

data. Hence, the variance of the returns relative to the new data is

M = (DTW−1D)−1

= [(τΣ)−1 + PTΩ−1P]−1 (3.14)

With this value M, the actual new variance (Σ) can be evaluated as [16]

Σ = Σ + M (3.15)

The final step in this process is to solve the mean-variance model by

using the posterior distribution of the Black-Litterman. Having the new vector

of expected returns and the covariance matrix, the new optimal portfolio can

be estimated using the standard mean-variance method

max xTµ− δ

2xTΣx (3.16)

The solution obtained using the first-order conditional is

Chapter 3. More Human-like Portfolio Optimization Approach 31

x∗ =1

δΣ−1µ (3.17)

(d) Idzorek

Idzorek describes an easy way to determine the level of trust by specifying

the confidence level of each view as a percentage [19]. This method is deemed

to be much more intuitive [14].

Another problem commonly found in the Black-Litterman model is the

determination of τ [44]. Idzorek calibrates the confidence of a view so that x/τ

ratio is equal to the variance of the portfolio view (pTΣp) [46], rendering the

scalar value of τ irrelevant. Idzorek still presents his formulas with τ , but it

can be removed in order to simplify the equations [14].

3.4 Experiments with Brazilian stocksOur process of composing a portfolio is divided in two stages: VDA and

Black-Litterman. In the first step, the investor must answer a series of questions

which will be used to create the views which, in turn, will be used in the Black-

Litterman to build the new portfolio. We created a methodology to construct

the view of the Black-Litterman model by using these questionnaires.

(a) Construction of the Views

Two different sets of questions were prepared. One of them is used to

identify what are the investor’s preferences regarding specific sectors of the

financial market, while the other aims at mapping the investor’s perspective

regarding the companies he/she intends to invest in.

The questionnaire about the sectors contains 3 questions. The first one

on how the domestic scenario is favorable to that sector, the second essentially

the same as the first but regarding the external scenario, and the last one on

the growth expectation for that sector. It was conceived simple, so it can be

answered by most people.

The other questionnaire, about the stocks, has 7 questions regarding risk,

reliability, expected growth, innovation, profitability, management, and com-

pany employees. It was also conceived to be as simple as possible, comprising

only a few questions.

In order to construct the views based on the answers given in the

questionnaires, we use the FIQ of the ZAPROS-III method. We consider the

FIQ as a rating through which we can quantify not only the classification of

stocks, but also how much one stock is better than another one. The FIQ has

Chapter 3. More Human-like Portfolio Optimization Approach 32

to be transformed into a standard for the views, and the values are normalized

between 0 and 1 to create an absolute view that represents the investor’s

perspective.

For questionnaires such as the one about sectors, in which an alternative

represents multiple stocks, we chose to equally divide the value attributed to

the sector among the stocks. For example, if the value of the sector is considered

to be 0.5 and we have two stocks, each one will have a value of 0.25.

Because confidence is a parameter that is somewhat complicated to

determine – even as a percentage –, we decided to insert one more question

in the questionnaires, in order to gauge how confident the participant is with

his/her answers, thus obtaining the confidence of the view. To discretize the

values, this question has four possible answers (very little confidence, little

confidence, reasonably confident and very confident), which are associated with

25, 50, 75, and 100 percent of confidence, respectively.

The last parameter of the view is the expected return. To have sufficient

impact on the portfolio, we chose 0.5% as its value. This value was chosen

based on the expected return of the assets and would be better calculated

automatically, but it was not possible to conceive a general formula which was

appropriate for any case.

(b) Results

To better understand how this methodology would behave in practice,

a test program was conceived to work with the Aranau [31] and Akutan

[14] frameworks. After the questionnaires have been filled out, the program

generates a graphical report showing the optimal portfolio and its details.

The Black-Litterman analytical resolution of the optimal portfolio has

some limitations: even while using a Lagrangian decomposition, like in Silva

et al. [47], the resulted formulation still cannot assure that the stocks’s

percentages are positive. Because of this limitation, the Jay Walters framework

has to be extended to solve the problems using the CPLEX2 solver.

We chose the 10 major companies negotiated in the Brazilian market3:

Petrobras, ItauUnibanco, Bradesco, Banco do Brasil, Vale, Itausa, Eletrobras,

Sid. Nacional, Cemig and Oi. For each of these companies, we chose the

stock with the highest negotiated volume to construct our portfolio and define

the corresponding sector. These companies operate in the following sectors:

electricity, financial, mining, oil, gas and biofuels, steel mill and metallurgy

and telecommunications.

2Version 12.4.0.03In 2012 according to Forbes

Chapter 3. More Human-like Portfolio Optimization Approach 33

After answering the questions, we obtained the FIQ values for the stocks

Table 3.1 and for the sectors Table 3.2. The lower the FIQ, the better

the alternative. Therefore, these values were normalized using the difference

between the maximum values of the companies.

Stock Sector FIQ Stock

Petrobras Oil, gas and biofuels 19

ItauUnibanco Financial 15

Bradesco Financial 26

Banco do Brasil Financial 19

Vale Mining 11

Itausa Financial 31

Eletrobras Electricity 39

Sid. Nacional Steel mill and Metallurgy 31

Cemig Electricity 31

Oi Telecommunications 46

Table 3.1: FIQ and sector of the stocks

The same normalization is done with the sector FIQ, but with the values

being distributed for all the stocks in the sector.

Sector FIQ Sector

Oil, gas and biofuels 12

Financial 1

Mining 7

Electricity 3

Steel mill and Metallurgy 8

Telecommunications 6

Table 3.2: FIQ of the sectors

The expected return were estimated as the mean of the daily returns

for February 2013, and these values are shown in Table 3.3. The returns vary

greatly, but this was not specifically for this month, as the Brazilian market

was experiencing some instability.

Chapter 3. More Human-like Portfolio Optimization Approach 34

Stock Exp. Ret.

Petrobras -0.6107

ItauUnibanco 0.1759

Bradesco -0.145

Banco do Brasil 0.4877

Vale -0.3962

Itausa 0.1924

Eletrobras -0.1139

Sid. Nacional -0.6274

Cemig 0.3786

Oi -0.5714

Table 3.3: The expected return of the stocks

Considering a confidence level of 75% for both the stocks and sector

questionnaires. Finally the views are composed by the confidence level, the

return and the normalized FIQ values for both the assets and the sectors,

which can be seen summarized in Table 3.4.

Stock View sector View stocks

Petrobras 0.14 0.00

ItauUnibanco 0.16 0.08

Bradesco 0.10 0.08

Banco do Brasil 0.14 0.08

Vale 0.18 0.14

Itausa 0.08 0.08

Eletrobras 0.04 0.13

Sid. Nacional 0.08 0.11

Cemig 0.08 0.13

Oi 0.00 0.17

Confidence 75% 75%

Return -0.0013 -0.00083

Table 3.4: A summary of the views data

Inputting the calculated views into the Black-Litterman, we obtain the

optimal portfolio of the Figure 3.3. To analyze how the portfolio changes, the

equilibrium portfolio is presented in Figure 3.4.

Chapter 3. More Human-like Portfolio Optimization Approach 35

Figure 3.3: The Black-Litterman portfolio with our views

Figure 3.4: Equilibrium portfolio

To analyze the sensibility of our method, we conducted some experiments,

as we shall see. However, we must emphasize that an improvement in the

qualification of an asset does not necessarily mean an increase of its percentage

in the optimal portfolio, as this variation also depends on the correlation and

on the assets’ return rates.

Answering the questionnaires with better expectations regarding the

growth, the risk, the innovation, the profitability and the employees of Sid.

Nacional, we obtained the portfolio in Figure 3.5. The participation of Sid.

Nacional’s stocks in the portfolio increased from 0.9% to 8.6%. The increment

was small because of the stock’s equilibrium return and high correlation with

Eletrobras.

Chapter 3. More Human-like Portfolio Optimization Approach 36

Figure 3.5: Result of the increase in the qualification of Sid. Nacional

The same thing happens if we increase the qualification of Oi, as seen in

Figure 3.6. In this case, the correlation between Oi and Bradesco is negative,

which explains why Bradesco’s percentage also increases.

Figure 3.6: Result of the increase in the qualification of Oi

We have similar behavior when we increase the qualifications of the

sectors, but in this case the change is less significant due to the return of

the sectors’ view and because the increase is distributed among all the sector’s

stocks.

To analyze how the resultant portfolio would perform in different situ-

ations it was simulated two different views considering the future return of the

stocks. It is assumed that the investor answers the questionnaires knowing the

asset that will have the highest return and that is the only asset that he/she

Chapter 3. More Human-like Portfolio Optimization Approach 37

wants to invest. In the first scenario we have the best possible outcome, i.e.

the investor guess was right and he/she invested on asset with highest return.

In the second scenario the asset behavior contrary of what the investor was

expecting, resulting on the worst portfolio return among all.

The performance was evaluated for a period of 6 months from February

to September of 2013. In Table 3.5 it is presented the returns of the stocks for

this period and Table 3.6 the returns for the sectors.

Table 3.5: stocks returns for the period

Asset Ret. (%)

Petrobras 4ItauUnibanco -6

Bradesco -13Banco do Brasil -7

Vale -7Itausa -7

Eletrobras -18Sid. Nacional -2

Cemig -3Oi -44

Table 3.6: Sectors returns for the period

Sector Ret. (%)

Oil, gas and biofuels 4Financial -8Mining -7

Electricity -10Steel mill and Metallurgy -2

Telecommunications -44

Considering these values for the first scenario the highest asset return

is Petrobras and the sector with the highest return is Oil, gas and biofuels.

For the second scenario the worst asset return is Oi and the sector with the

worst return is Telecommunications. Resultant portfolios obtain by answering

the questionnaires considering those scenarios are presented in Figure 3.7 and

3.8.

In Table 3.7 also compare these two portfolios with the Market Portfolio

(optimized portfolio with the best Sharpe Ratio) and the portfolio generated

before (Previous).

Analyzing the worst scenario it is evident the problem of allocating the

portfolio entirely in an active disregarding the risk. If the investor had invested

everything on Oi he/she would have lost 44% of its initial investment, using

Chapter 3. More Human-like Portfolio Optimization Approach 38

Figure 3.7: Result of right scenario, where the investor guess is right

Figure 3.8: Result of wrong scenario, where the investor guess is wrong

Table 3.7: Return for the different scenarios and the Market Portfolio

Portfolio Ret. (%)

Previous 93.04Market Portfolio 95.19Right Scenario 97.46Wrong Scenario 82.35

the methodology proposed in this article that loss would decrease to 18%.

However, a gain that would have invested in investor Petrobras decreases from

4% to -3%. These results indicate a decreased risk of the portfolio.

4Dynamic Asset Allocation Under Uncer-tainty

In real life applications, we deal with decision-making processes marked

by uncertainty, i.e., with choices having to be made considering future realiza-

tions of risk factors. Stochastic Programming aims at finding optimal solutions

for such cases involving uncertain data. These decisions also have to be con-

tinuously and periodically re-evaluated, which is why real life applications are

often modeled as multistage stochastic problems.

However, there are many challenges in modeling and solving this type of

application in order to avoid the curse of dimensionality, which is the major

obstacle for multistage stochastic decisions. In fact, in several cases analysts

have to deal with a very large number of possible outcomes at each stage and

the scenario trees grow exponentially with the number of stages. Thus, as the

problems grow, the number of scenarios can easily become enormous.

4.1 IntroductionThere are two common types of methods that can be implemented to

solve multistage stochastic problems. Methods for scenario decomposition, like

progressive hedging and L-shaped (nested Benders); and sampling-based de-

composition methods, such as Stochastic Dynamic Dual Programming (SDDP)

[48], Abridged Nested Decomposition (AND) [49] and Convergent Cutting-

Plane and Partial-Sampling Algorithm (CUPPS) [50].

It is important to highlight that in most cases only sampling-based

methods can efficiently solve problems with a large number of stages, as even

for modern processing power other methods still require a huge computational

effort. Hence, when dealing with large instances, sampling-based algorithmis

are the most appropriate approach.

Pereira and Pinto [48] introduces the SDDP method, through a scenario

sampling process it adds new constraints on the problem using the dual

optimal solution. To overcome the curse of dimensionality, it assumes temporal

independence to separate cost-to-go functions by stage. Sometimes, however, it

is not reasonable to assume stagewise independence for some kind of problems.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 40

Therefore, if the problem presents some level of temporal dependence, the

challenge is to modify the SDDP methodology keeping the cost-to-go function

independent by stage.

Some problems with temporal dependence in the right-hand side, such as

the hydrothermal scheduling, are treatable augmenting the state space [51]. In

the hydrothermal model, for example, the rivers’ inflows are time-dependent

and, in order to create the independence between stages, the proposed solution

is to use the inflow as state variable [52] [53]. To represent this linear

correlation, it is also necessary to change the right-hand side accordingly.

However, this approach is specific for problems with right-hand side linear

dependence.

Mo et al. [54] and Philpott and Matos [55] have shown that when the

relationships between stages are nonlinear, the problem can also be modified

to maintain stagewise independence. Using the Markov chains to model

the nonlinear temporal dependence, it is also possible to preserve stagewise

independence between cost-to-go functions. As in the other methods mentioned

above, the downside of these approaches is to increase the state space, thus

raising the computational effort needed.

Furthermore, it is well-known that in financial time series returns also

have a nonlinear dependence [56, 57]. Portfolio optimization problems can be

modeled as multistage stochastic problems in which for each stage a decision

has to be made regarding where to invest. Therefore, multistage problems

concerning asset allocation have temporal dependence on the left-hand side

and are usually solved using scenario trees like in Valladao et al. [58]. However,

not all the uncertainty regarding a problem can be represented using scenario

trees, because it would become computationally untreatable.

When Dantzig and Infanger [59] researched this problem in 1993, they

used classical Benders decomposition to solve the multistage portfolio problem.

Nonetheless, their method relied on multidimensional integration, and the issue

of dependence between stages was stated but not resolved.

Assuming temporal independence and no transactional cost, the optimal

multistage asset allocation solution is obtained by a myopic policy [60],

allowing such problems to be easily solved by solving a 2-stage problem.

However, when confronted with the reality of transaction costs and temporal

dependence, the proof of optimal myopic policy is not valid.

Our proposal in this chapter is to solve practical multistage asset alloca-

tion problems with realistic assumptions. Modeling the temporal dependence

as a Hidden Markov Model (HMM), using Gaussian mixture, where the state

likelihood depends on historical returns.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 41

In SDDP, we will create a worth function for each Markov state and

stage, with the states probabilities at each stage depending on past returns.

The downside of this approach is that it increases the size of the state

space. Normally, for financial series the states of the Markovian process are

concealed. Consequently, the HMM – a well-established technique – has been

used to predict returns and determine the states probabilities. Using the worth

functions for each Markovian states this proposed methodology will solve an

approximation of real problem.

Similar approach was used in Mo et al. [54] Philpott and Matos [55],

although their approach are different, the dependence is on the right-hand

side, while in the asset allocation problem it is on the left-hand side, and they

have observable states.

In these dynamic decision problems, like multistage asset allocation,

there are much uncertainty about the future realizations making necessary

to incorporate some form of risk aversion in the optimization model. Several

risk measures have been used for portfolio optimization [42], the Conditional

Value at Risk (CVaR) has recently become widely used because it is a coherent

risk measure [61,62] [63] [51].

It is very important to use a time consistency model with a clear

interpretation of the objective function. To overcome skepticism and earn

the investors’ trust, it is necessary that they understand what is the main

objective of the optimized portfolio. Time consistency is also a concern, since

inconsistency may not take risk aversion into account. Therefore, we are going

to adopt a recursive model based on convex combination of the expected

value and CVaR, as it ensures temporal consistency and can be interpreted

as certainty equivalent of the fund.

Section 4.2 describes a brief review of the SDDP with risk neutral and risk

averse proposals, in Section 4.3 we present the SDDP methodology modified

to asset allocation, with different assumptions, our proposal the H2SDDP and

the ambiguity aversion approach. Finally, Section 4.4 brings the experiments

in portfolio selection to analyze how the methods will perform.

4.2 Stochastic Dynamic Dual ProgrammingStochastic Dynamic Dual Programming (SDDP) is an extension of

the Benders decomposition [64] that overcomes the curse of dimensionality,

common in multistage stochastic problems, by creating for each stage an

approximation of the cost-to-go function using a set of linear functions (cutting

planes) [48].

Chapter 4. Dynamic Asset Allocation Under Uncertainty 42

In the literature we can find other multistage stochastic programming

methods that also uses sample average approximation [50] [49], all these meth-

ods successively sample the distribution, constructing better approximations of

the cost-to-go function. What distinguishes SDDP from the other methods is

how the cutting planes are evaluated, using the dual solution, making possible

to share cuts between solution of the same stage.

SDDP, and most of the sampling-based methods, assumes stagewise

independence to avoid having multiple cuts by stage. Considering stagewise

independence, the cutting planes can be shared between solutions from the

same stage. Thus only one approximation of the cost-to-go function has to be

constructed at each stage.

The SDDP algorithm is divide into two main parts, a simulation forward

in time and a backward in time recursion. It generates trials solutions xtj in

forward step and in backward step uses this solutions to compute the cutting

planes for cost-to-go functions.

More specifically, forward step sample m ∈ 1, . . . ,M scenarios to

construct trial solutions xmt chosen from Nt random realizations for every stage

t. For each of these scenarios and stages it is evaluated the optimal solution

using the current approximation of the cost-to-go functions. This procedure

has this name because it goes forward in time taking samples from 1 to T

stages.

Backward step goes in the opposite time direction, from T to 1, taking

advantage of subsequent evaluated cuts. In backward step, the trial solutions

obtained in forward step are used as the solution for previous stage xt−1. For

each stage and sample it solves a problem to find the optimal value xt. These

results are used to calculate the cutting plane of the stages and then added to

the corresponding stage’s set of cutting planes.

(a) Risk Neutral SDDP

The following section is dedicated to present the dynamic programming

equations for the original SDDP, keeping in mind the assumption of stagewise

independence. The notation adopted herein is similar to that of Shapiro in

[60]. SDDP constructs a Sample Average Approximation (SAA) of the “true”

problem by sampling scenarios, typically using Monte Carlo. The generic risk

neutral formulation for multistage stochastic problems is

maxA1x1=b1

x1≥0

c>1 x1+E

[max

A2x2=b2−B2x1x2≥0

c>2 x2+...+E

[max

ATxT =bT−BTxT−1xT≥0

c>T xT

∣∣∣ξT−1

]...

∣∣∣∣ξ1]

(4.1)

Chapter 4. Dynamic Asset Allocation Under Uncertainty 43

The number of stages is T and the number of conditional discrete

realization in each stages is Nt. In this problem, a realization is

ξt(s) = ct(s),At(s),Bt(s),bt(s) (4.2)

To reduce notation, in this formula and throughout the text ξt(s) has

been simplify, leaving ξts = cts,Ats,Bts,bts. Assuming independent samples

with ξt ∈ ξt,1, . . . , ξt,Nt, ∀t ∈ 2, . . . , T, that means ξt does not depend on

ξ1, . . . , ξt−1 and that the cost-to-go functions Qt+1(xt, ξt+1) is independent

from past data. The multistage stochastic dynamic programming equation is

Qt(xt−1, ξts) = maxxt

c>tsxt +Qt+1(xt) (4.3)

s.t. Atsxt = bts −Btsxt−1 : πts

xt ≥ 0

where πts is the vector of dual variables.

In this case, the cost-to-go function is

Qt+1(xt) =1

Nt

Nt∑s=1

Qt+1(xt, ξt+1,s), ∀t ∈ H (4.4)

In the last stage QT (·) ≡ 0. The cost-to-go function Qt+1(xt) shall be

approximated by a set of cutting planes It

Qt(xt−1) = maxl∈ItQt(xt−1,l) + g>tl (xt−1 − xt−1,l), ∀t ∈ H (4.5)

where Qt(xt−1,l) = 1Nt

∑Nt

s=1 Qt(xt−1,l, ξts) and gtl = − 1Nt

∑Nt

s=1 πtslBts.

Using the dynamic programming equations, the SDDP algorithm can

be explained in further detail. In forward step, it uses Qt+1(xt) with current

collections of cutting planes, to find the trials xm1 , · · · ,xmT for backward step.

Backward part goes from T to 1, using the trial solutions as xt−1 to add new

cuts by evaluating Qt(xt−1,l) and gtl. The number of scenarios sampled in

forward will determine how many cuts will be added for iteration.

An estimate of an lower bound is evaluated during the forward procedure

by multiplying the trial solutions by the cost. The upper bound is obtained by

solving the first stages’ problem, using the sets of cutting planes Q2(·).

(b) Time Consistent Risk Averse Model

This section presents the formulation for the risk averse model for

multistage asset allocation problems, using CVaR as risk measure and assum-

ing independence between stages. It is important to notice that the definition

Chapter 4. Dynamic Asset Allocation Under Uncertainty 44

of Value-at-risk for gain distributions [63] differs from the original [65], when

the domain is the final net worth(gain). To differentiate both definitions, we

will call the Value-at-risk for gain distributions VaR and the traditional Value-

at-risk for net worth VaR, analogously for CVaR and CVaR. As this section

will cover gain distributions, it will use CVaR, as in [63].

Conditional value at risk

Standard deviation is the most commonly used measure of dispersion

for asset allocation, however, there are other popular risk measures, such

as Absolute Deviation and Mean-Absolute Moment, as well as downside

measures like semivariance, Value at Risk (VaR) and Conditional Value at

Risk (CVaR), [1]. While dispersion measures attribute an equal weight to the

deviations below and above the mean, for downside risk measure the most

important part is on realizations worse than the portfolio mean return. The

downside risk measure suggests that the left-hand side of a return distribution

gives information about risk, while the right-hand side contains investment

opportunities [66].

One of the preferred downside measures is VaR, vastly used by financial

institutions to report on the exposure of their portfolios to risk. VaR is the

maximum portfolio loss within a certain confidence level α, normally 95% or

99% 1. The popularity of this measure is due to its simple interpretation as the

maximum reasonable loss for a given portfolio. Formally, for a random variable

Z VaR is defined as2

V aRα(Z) = infl : P(ω ∈ Ω | −Z(ω) ≥ l) ≤ (1− α) (4.6)

Despite the popularity of VaR, this risk measure has several undesirable

properties. For one, it does not consider extreme events, changes on the tail

distribution will not have any impact on VaR. It is not subadditive, and more

diversification may lead to an increase in the portfolio risk, what also creates a

non convex function, making it unattractive for optimization. These issues with

VaR motivated the creation of a set of desired properties for risk measures, i.e.

what properties they should have to become coherent risk measures [65].

Therefore, using Z as the set of possible gains, for a risk measure

ρ : Z → R to be coherent it must satisfy the following axioms (for Z and

Z ′ ∈ Z):

1. Convexity, ρ(lZ + (1− l)Z ′) ≤ lρ(Z) + (1− l)ρ(Z ′) l ∈ [0, 1];

1This definition differs from [63], where the (1 − α) is the confidence level and α is asmall number

2As we are using gains Wt, we will use the original definition of VaR [65] when the domainis gain and not loss.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 45

2. Monotonicity, if Z ≤ Z ′ then ρ(Z) ≤ ρ(Z ′);

3. Translation equivalence, if a ∈ R then ρ(Z + a) = ρ(Z) + a;

4. Positive homogeneity, if l > 0 then ρ(lZ) = lρ(Z).

It is common to find in the literature studies defining the subadditivity

property instead of the convexity, because subadditivity and positive homo-

geneity are sufficient to ensure that ρ(·) is convex.

Over the last decade, a special effort has been made to use and find coher-

ent risk measures, CVaR being one example of solution found for this purpose.

Also known as expected tail loss (ETL), it is the conditional expectation of

losses beyond the VaR, and the mean of the distribution tail on the right side

of the VaR. It can also be simply explained as the expected loss when the VaR

is exceeded. CVaR is defined for a loss distribution(-Z)

CVaRα(Z) = VaRα(Z) + (1− α)−1E[−Z − VaRα(Z)]+ (4.7)

Initially proposed by Rockafellar and Uryasev [62] for portfolio optimiz-

ation, the advantages brought by the CVaR, often proposed as a substitute for

the VaR, have led to many studies and applications [61, 63, 67]. It was lately

used by Shapiro [63] in conjunction with SDDP to propose and solve a risk

aversion multistage stochastic problem.

In definition (4.7), it is important to note that CVaR is an upper bound

for VaR, i.e. CVaR is always equal or greater than VaR. As a result, one

solution with constraint CV aRα(Z) ≤ κ will remain feasible for V aRα(Z) ≤ κ.

Figure 4.1, illustrating the concept of VaR and CVaR for loss distributions.

Figure 4.1: -CVaR and -VaR for gain distribution

Chapter 4. Dynamic Asset Allocation Under Uncertainty 46

As a coherent risk measure, the CVaR should be an appropriate way

to assess the risk. However, we want to stress the fact that if one chooses a

dynamic model, time consistency should be a concern since an inconsistent

policy may not consider aversion risk [68].

In a maximization problem it can also be used an alternative definition

using -CVaR to evaluate the equation above, as follows

−CVaRα(Z) = supu∈R

u− E[(Z − u)−]

1− α

(4.8)

where x− = −min(x, 0).

(c) Stopping Criteria

For general deterministic problems, as well as for deterministic Benders

decomposition, the optimal solution is achieved when the lower bound and

the upper bound are the same. However, the SDDP problem is SAA of the

“true” problem, thus the lower and upper bound are an approximation of the

actual value for the real “true” problem, and one has to consider the confidence

interval of this values to compare them.

Although the finite convergence of SDDP and other sampling-based

methods has been proven [63, 69], the computational time required for the

algorithm to actually converge could be considerable. The traditional stopping

criterion proposed by Pereira e Pinto [53] consists in stopping SDDP when

the upper bound get into the lower bound’s confidence interval. As exposed

by Shapiro 2011 [63], this criterion does not seem to make much sense and

it would be more logical if the algorithm stopped when the gap between the

upper limit of the lower bound and the upper bound became small enough

(< ε). However, this gap can be very large and normally does not get narrow

enough within a reasonable computation time [70].

There has been debate around what should be the stopping criteria

of SDDP. Several other different stopping criteria have been proposed in

the literature, but to this date none has been widely accepted as being the

best one. Additionally, finding the lower bound for the multistage stochastic

maximization problems that uses nested CVaR [63] is not easy. In spite of

some promising attempts [71], as seen above, there is no guarantee that by

using a stopping criteria with lower bound the solution of SDDP will converge.

Actually, in practical problems, achieving convergence is very time-consuming,

thus most of the time the SDDP has to be stopped before finding the optimal

solution. Further debate on the convergence of SDDP can be found in [63,72].

Despite all the discussion around the convergence of SDDP, the most

Chapter 4. Dynamic Asset Allocation Under Uncertainty 47

straightforward method to stop SDDP with nested CVaR is the stabilization

of the upper bound. As pointed out by Shapiro [63], the stabilization of the

upper bound does not mean that SDDP is close to the optimal solution.

However, during this thesis, when stabilization was achieved, even by changing

the number of trials and leaving the algorithm running for many days, the

upper bound could not be changed. Thus, given the fact that none of the

methods has guaranteed convergence, we think that the stabilization of the

upper bound represents the better trade off between the computational time

and proximity to the optimal solution.

(d) Sampling Scenarios

The idea to represent the random process through samples was used

by different authors over the years. Sample Average Approximation (SAA)

consists in representing a distribution by random realizations sampled from

“true” distributions. Therefore, the “true” optimization problem is replaced

by its SAA version. The method traditionally used in SAA is the Monte Carlo,

that randomly selects independent and identically distributed (i.i.d.) samples,

all realizations having an equal probability of being selected.

In practical situations, sampling the scenarios with Monte Carlo requires

a large number of samples in order to keep deviations small. Nonetheless,

there are techniques that can be used to reduce the variance, like Importance

Sampling and Latin Hypercube Sampling.

Importance sampling having been created to be a variance reduction

technique [73], but is also particularly useful when it is impossible to sample

from the real distribution, it allows to study and analyze a distribution using

samples taken from another distribution.

Assuming that the likelihood ratio function f(xi)g(xi)

is well defined, if there

is some z ∈ Rd where f(z) = 0, then g(x) = 0, the expected value of a random

variable X ∼ h can be calculated as

E(h(X)) =

∫Dh(x)f(x)dx =

∫D

h(x)f(x)

g(x)g(x)dx = Eg

(h(X)f(X)

g(X)

)(4.9)

For a given set of i.i.d. samples xi drawn according to a probability

density g(X) the importance sampling estimator of µ = Eg(h(X)) is

µg =1

n

n∑i=1

h(xi)f(xi)

g(xi), xi ∼ g (4.10)

The importance sampling estimator µg is an unbiased estimator of µ. By

sampling from distribution probability density function g(·) and making an

Chapter 4. Dynamic Asset Allocation Under Uncertainty 48

adjustment using the likelihood ratio f(xi)g(xi)

, it is also possible to obtain samples

from the original or nominal distribution h.

This approach in SDDP may be used, for example, for a cost-to-go func-

tion Qt(xt−1, ξts), while an importance sampling estimator can be evaluated

given realizations different form ξ. However, as indicated by Shapiro in [60],

the importance sampling method has some issues, it is very sensitive to the

choice of p.d.f., for example, and its instability is notorious, even for small

perturbations in g(·).Latin Hypercube Sampling, another variance reduction technique, uses

probability dense inverse function to sample the distribution more evenly.

Initially proposed by Mckay et al. 1979 [74], it is a stratified sampling method, a

K-dimensional extension of Latin square sampling. Latin Hypercube Sampling

divide the p.d.f. in intervals with the same probability and collect one sample

from each interval. Controlling how random samples are generated, the method

spreads the samples more evenly throughout the distribution.

It is possible to observe on Figure 4.2 how Latin Hypercube Sampling

method performs on two uniform distributions X ∼ U(0, 1) and Y ∼ U(0, 1).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

X

Y

Figure 4.2: Example of Latin Hypercube Sampling for uniform distributionwith two dimensions

To illustrate how Latin Hypercube Sampling and Monte Carlo behave,

two examples were designed, which can be seen in Figure 4.3 and Figure 4.4. For

the first example, five hundred iterations were performed, sampling 30 points of

normal distribution N ∼ (0, 1) by using Monte Carlo and also Latin Hypercube

Sampling. The resulting figures present the estimated normal distribution for

each one of the five hundred iterations.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 49

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

f(x)

Figure 4.3: Sampling withMonte Carlo

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

f(x)

Figure 4.4: Sampling withLatin Hypercube Sampling

On the figure it is clear that the variance of the Monte Carlo is larger

than the Latin Hypercube Sampling. Latin Hypercube Sampling also requires

a reasonable number of examples, mainly to better represent the distribution

tails. The intervals on the tails are bigger, as we can see in Figure 4.5, thus more

samples would be necessary to better represent this part of the distribution.

−3 −2 −1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Figure 4.5: LHS interval for a normal distribution N ∼ (0, 1)

4.3 Stochastic Dynamic Programming for AssetAllocation

In this section we will explain how to adapt SDDP for multistage asset

allocation. First, we will begin with the risk-neutral formulation and risk averse

Chapter 4. Dynamic Asset Allocation Under Uncertainty 50

formulation, followed by the time independent model, transactional costs with

SDDP and, finally, by our proposal to treat temporal dependence.

Supposing an investor wishes to construct a portfolio dividing a given

amount of cash among the assets xti, ∀i ∈ A each of which has an uncertain

return rti(s) for time t. Each stage brings the need to reallocate the portfolio,

considering the profits obtained in the previous stage Wt = (1 + rt)>xt−1.

Considering an initial capital W1, and not allowing a short sell (i.e. x ≥ 0),

the risk-neutral portfolio multistage stochastic problem is stated as

max1>x1=W1

x1≥0

E

[max

1>x2=W2x2≥0

E[

max1>xt−1=WT−2

xt−1≥0

E[WT

∣∣r[T−1]

]. . .

∣∣∣r1

]](4.11)

The realization of the investment made in t is accounted for in the following

stage t + 1. The optimal solution for this risk-neutral formulation would be

to allocate everything in the asset with the highest expected profit. However,

it is a known fact that investing everything in a single asset is not very wise.

Neglecting risk in the optimization problem can lead to big losses, especially

in fall trends.

Using ψ(·)t function it is possible to add risk aversion formulation for

asset allocation

max1>x1=W1

x1≥0

ψ1

[max

1>x2=W2x2≥0

ψ2

[max

1>xt−1=WT−2xt−1≥0

ψT−2

[WT

∣∣r[T−1]

]. . .

∣∣∣r1

]](4.12)

The function ψ(·)t is the convex combination combination of the expect value

and the CVaR, as it is a maximization problem. Formally, the function ψt(·)is

ψt(Z|ξt) = (1− λ)E[Z|ξt] + λ(−CVaRα[Z|ξt]) (4.13)

(a) Myopic policy: No transaction costs and temporal in-dependence

For the special case when the multistage portfolio selection does not

have transactional cost and no temporal dependence the optimal solution can

be easily found using a myopic policy. In the appendix A.2 it is presented the

myopic prove for a coherent risk measure [60].

The myopic policy, that aims on maximizing one-step reward ignoring

the impact on future rewards, is optimal. However, this policy is optimal in a

very specify situation. In a more realist situation, with transactional costs, the

myopic policy is not optimal.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 51

(b) SDDP for asset allocation: Transaction costs and tem-poral independence

In this section, the asset allocation problem will be formulated with xt

in the state space and risk aversion with CVaR, in order to be used in SDDP

and to later include transactional costs. However, this model still disregards

the temporal dependence.

According to Rudloff et al. [68], the nested CVaR is the certainty

equivalent of the portfolio and, consequently, the worth of the portfolio.

Therefore, for the multistage asset allocation problem the cost-to-go is actually

the worth function. The dynamic programming equation can be defined using

the risk averse multistage formulation for asset allocation (4.12) and the

function ψ(·)(4.13). The worth function is

Qt(xt−1, rt) = maxxt

r>t xt−1 +Qt+1(xt) (4.14)

s.t. 1xt = (rt + 1)>xt−1

xt ≥ 0

and the expected recourse function, in this case, the portfolio expected worth

function is

Qt+1(xt) = (1− λ)1

Nt

Nt∑s=1

Qt+1(xt, rt+1(s))+ (4.15)

λ

(Qt+1(xt, rt+1(τ))− 1

(1− α)Nt

Nt∑s=1

[Qt+1(xt, rt+1(s))−Qt+1(xt, rt+1(τ))]−

)

Is important to emphasize that the Qt+1(xt, rt+1(τ)) is the -VaR of

Qt+1(xt, rt+1(1)), . . . , Qt+1(xt, rt+1(Nt)), i.e the right or positive α quantile.

Herein the -CVaR, and consequently -VaR, will be evaluated using the dual

representation of the CVaR that is a fractional knapsack problem [61,65].

−CV aR(X) = minvs

Nt∑s=1

Qt+1(xt, rt+1(s))× vs (4.16)

s.t.Nt∑s=1

vs = 1

vs ≤1

(1− α)Nt

∀s ∈ 1, . . . , Nt

vs ≥ 0 ∀s ∈ 1, . . . , Nt

To determine the cut, it is necessary to evaluate the subgradient of the

Chapter 4. Dynamic Asset Allocation Under Uncertainty 52

worth function Qt(xt−1), it was used -CVaR. With γts being the subgradient

of Qt(x∗t−1) at point x∗t−1, the subgradient of the expected recourse function is

gt = ∇Qt(x∗t−1) = (1− λ)1

Nt

Nt∑s=1

γts + λ

(γtτ −

Nt∑s=1

ηts

)(4.17)

where using the weights vs of (4.16) we have ηts∀s ∈ 1, . . . , Nt

ηts =

0 if Qt(xt−1, rt(s)) > Qt(xt−1, rt(τ))

−(γt+1,s − γt+1,τ )vs if Qt(xt−1, rt(s)) ≤ Qt(xt−1, rt(τ))(4.18)

and the subgradient of Qt(xt) changes to

γt = rt + (1+ rt)>π∗ (4.19)

It must be kept in mind that traditional SDDP is a minimization problem,

however, this work maximizes the objective function, thus the solution of

Qt(xt−1, rt) is an upper bound to the model presented here.

Transactional costs

Most of the early asset allocation methods did not consider the transac-

tional costs on purchases or sales of assets. When the costs of each transaction

are neglected, the gain apparently obtained can be misleading. Depending on

the frequency of the transactions and costs, an estimated positive return could,

in reality, became loss. Moreover, costs stimulate the maintenance of the se-

lected portfolio in the previous period, because it is an explicit regularization

of portfolio movements. It is a L1 regularization [75], since the sum of the

absolute differences, the changes on the portfolio, is penalized on the object-

ive function. It will be seen later in the experiments section this behavior in

practice.

Introducing d+ti and d−ti to represent how much from each asset i is bought

and sold. Also, the variable c that is the proportional cost, selling will bring

an income (1− c)d−ti , and buying will mean a cost (1 + c)d+ti to current wealth

(1+ rt)>xt−1. The problem with transactional cost can be formulated as

Qt(xt−1, rt) =

maxxt,d

+t ,d−t

r>t xt−1 −∑i∈A

c(d+ti + d−ti) +Qt+1(xt) (4.20)

s. t. 1>xt + c∑i∈A

(d+ti + d−ti) = (1+ rt)

>xt−1 (4.21)

xti − d+ti + d−ti = (1 + rti)xt−1,i, ∀i ∈ A \ 1 (4.22)

d+t ,d

−t ,xt ≥ 0 (4.23)

Chapter 4. Dynamic Asset Allocation Under Uncertainty 53

Notice that there is no transactional cost in the purchase or sale of risk-

free asset. Furthermore, in this model the complementarity of the simultaneous

purchase and sale of a given asset is guaranteed, i.e. d+tid−ti = 0. If there

were simultaneous purchase and sale operations of an asset, that could not

be the best solution as it would be possible to obtain a better return without

interfering with xti.

The subgradient of the worth function for SDDP also changes. In this

context there are additionally A dual variables, represented by the vector

π2 ∈ RA associated with constraint (4.22), then the subgradient of Qt(xt)

becomeγt = rt + (1+ rt)

>π∗1 + (1+ rt)>π∗2 (4.24)

with π1 representing the dual variables of constraint (4.21). Furthermore, the

optimal solution of this new problem cannot be found using myopic policy,

now it cannot ignore future implications caused by the current decision.

(c) H2SDDP for allocation: Transaction costs and temporaldependence

In previous sections it was shown that when the multistage asset alloc-

ation problem has no time dependency the optimal solution can be achieved

using a myopic policy. When transactional costs are added to the model this

myopic optimal behavior does not hold, under these circumstances the tradi-

tional SDDP can be used to find the optimal solution. However, the stagewise

assumption of traditional SDDP is an issue when modeling temporal depend-

ence problems. In this section we will discuss some approaches to model time

dependency and how to modify SDDP for this purpose.

In many applications a more general problem has to be considered, when

there is time dependency between the stages. The worth function becomes

dependent on the historical achievements of the random variable Qt+1(xt, r[t]).

Thus, we have a worth function (future cost function on original SDDP) for

each node in the stochastic tree, and therefore is so common to use L-shaped

methods. The hydrothermal scheduling is one of the problems that typically

has time dependency. This dependence usually occurs in inflows, on the right-

hand side of the problem. The inflows are modeled as an AR(1) autoregressive

of order one, that isbt = φtbt−1 + εt (4.25)

Changing the stochastic process to ξt = (ct,Bt,At, εt) and increasing

the state space to (xt,bt), we can define the generic problem as well as solving

it using the similar SDDP method used by independent case, problem (4.3),

Chapter 4. Dynamic Asset Allocation Under Uncertainty 54

with the cost-to-go function as

Q(xt−1,bt−1, ξt) = maxxt

c>t xt +Qt+1(xt) (4.26)

s. t. Atxt = −Btxt−1 + φtbt−1 + εt

xt ≥ 0

Furthermore, in hydrothermal scheduling there are situations where the

inflows are best mapped as nonlinear functions, for example, as a Markov

process. There are some advantages in mapping the time dependency as a

Markov process, in addition to encapsulate all the historical information it

also incorporate autocorrelated dependency, this is extremely important since

seasonality is present in the inflows data [76].

Using Markovian process to represent the inflows probability distribution,

it is also possible to preserve stagewise independence between cost-to-go

functions. This approach was first develop in [54], and later adapted in [55]

for the New Zealand electricity system and similar methodolodgy was applied

to Energy contracts management of Liquefied Natural Gas in [77]. All these

approaches use deterministic states and has dependency on the right-hand side.

H2SDDP

Undoubtedly, the methodology developed by Pereira and Pinto [48],

SDDP, has great advantages, however, the assumption of stagewise independ-

ence is not realistic for some problems. When dealing with with temporal

dependence, the methodology must go through some adaptation in order to

be able to solve this type of problem.

In the alternative L-shaped method, there is one worth function for every

node of the scenario tree, each node has a unique history and the worth

function depends on the historical return. Therefore, this technique has greater

computational complexity since the expected worth function has to consider

the whole previous realization of the scenario.

The solution proposed here for multistage asset allocation is the Hidden

States Stochastic Dynamic Programming (H2SDDP), that can be described as

a combination of SDDP and L-shaped procedures. In the H2SDDP method,

the market is classified and separated by states, so only the probability of each

state depends on the past returns. The method can be seen as a generalization

of SDDP and L-Shaped, if the number of states is equal to the number of nodes

in the stochastic tree, the model will behave like L-Shaped; otherwise, if the

number of states is one, it will act similarly to SDDP.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 55

The foremost difference between the three methods (SDDP, L-Shaped

and H2SDDP) is the trade-off between how historical information is considered

in each node of the stochastic tree and the computational complexity involved.

With H2SDDP method we expected to achieve the best compromise in this

regard, as it creates a small increase in complexity while being able to deal

with problems with temporal dependence within a reasonable computational

time.

In this work the temporal dependence will be expressed as states of hid-

den Markovian process and the asset return will be modeled as a multivariate

mixture of Gaussians. There are other ways to separate financial time series

in states, the HMM was chosen because it is more suitable for necessities in-

volved in this circumstances. Indeed, the method is especially appropriate for

this application because it can be used to estimate the states likelihood and

to sample the return distributions given the states.

Many traditional financial applications used the normal distributions

with stationary parameters to model the log-returns distributions. Nowadays,

it is well known that this does not reproduce some important stylized facts or

well known properties of the financial time series.

Fama [78] study analyses the behavior of stock market prices and presen-

ted some evidences that contradicts the assumption of normality. Granger and

Ding [79] presented some of the stylized facts of properties for a variety of

log return series. Ryden et al. [80] shown that modeling the financial time

series as a Gaussian mixture according to the states of an unobserved Markov

chain, the HMM model, reproduces most of the stylized facts for daily return

series demonstrated by Granger and Ding [79].Thus, HMM is frequently used

to model return series [80–83].

Figure 4.6 is an example of a simple a HMM for stock market, which

uses, for the sake of simplicity, discrete emission distributions. The objective

is to determine the market state Bull, Bear, Neutral given the return status

of the current day stable, down, up. Observe that, given an emission, it is not

possible to determine, with certainty, what is the state; and given a state and

using the state emission distribution, we can obtain a realization.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 56

Neutral

Bull Bear

downstableup downstable up

down

stable

up

0.10.2

0.7 0.60.30.1

0.2

0.6

0.2

0.2 0.2

0.2

0.1

0.10.1

0.6

0.7 0.8

Figure 4.6: An example of HMM for stock market

But in this application the return of each asset will depend on the state

of the HMM, Figure 4.7 is an example on how return can be represented by

HMM as a mixture of Gaussians.

Figure 4.7: An example of a Gaussian mixture model.

Since temporal dependence of the return series will be modeled as a

HMM, the SDDP’s worth function will be separated by state, creating a

stagewise independence. Therefore, there is only one worth function for each

Chapter 4. Dynamic Asset Allocation Under Uncertainty 57

stage that depends on a specific market state. These functions do not depend on

the previous returns, only the probability of the Markov states is conditional to

historical returns. The downside of this approach is increasing computational

effort, since it has more worth functions, one per state.

Let us first assume that at time t the market state k is known, then we

solve the following problem

Ut(xt|k) = ψt[utk(xt, rt+1)|k] (4.27)

utk(xt, rt+1(s)) = rt+1(s)>xt + maxxt+1∈X(xt,rt+1(s))

K∑j=1

Ut+1(xt+1|j)P (j|k)

where Ut(xt|k) is the worth function given the state k and the transition

probability from state k to j is P (j|k).

In practice, since the state k is unknown, we need to solve problem (4.27)

using the past observed returns r[t] to estimate the states probabilities. To

simulate the system and find the optimal policy we solve the following problem

RPt (xt−1, r[t]) = r>t xt−1 + max

xt∈X(xt−1,rt)

K∑k=1

Ut(xt|k)P (k|r[t]) (4.28)

where P (k|r[t]) is the likelihood of the state k estimated by the forward-

backward algorithm [84] given the past returns r[t].

The worth function Qt+1(xt) is approximated by a set of cutting planes

ItUt(xt|k) = max

l∈Itutk(xt−1,l) + g>tl (xt−1 − xt−1,l), ∀t ∈ H (4.29)

where utk(xt−1,l) = 1Nt

∑Nt

s=1 utk(xt−1,l, rt(s)) and gtl analogously to

equation 4.17 with the subgradient equal to 4.24.

To solve the problem (4.28), it is necessary to know each state’s prob-

ability, in order to weigh the worth function of the next stage. Our approach

with HMM uses past returns to estimate the present probability of each mar-

ket state in forward step. With the states’ probabilities, the objective function

can be evaluate by combining the state probability with the respective worth

function. Also, the states’ probabilities of each stage determine the joint dis-

tributions that are used to establish the sequence of observed returns.

In backward step, each state emission distribution is considered and

one or more cuts are added to each respective state. In other words, when

realizations from a given state are used, the cut will be added to the set of

cuts of that same state. Also, as in problem (4.28), the worth function is

conditional on the state, meaning there will be K functions per stage. Using

the dynamic programming equations above, the models used in forward and

Chapter 4. Dynamic Asset Allocation Under Uncertainty 58

backward steps are (4.30) and (4.31), respectively, with transactional costs.

xt = arg maxxt,d

+t ,d−t

−∑i

c(d+ti + d−ti) +

K∑k=1

Ut(xt|k)P (k|r[1,t]) (4.30)

s. t. 1>xt +∑i

c(d+ti + d−ti) = (1+ rt)

>xt−1

xti − d+ti + d−ti = (1 + rti)xt−1,i, ∀i ∈ A \ 1

xt,d+t ,d

−t ≥ 0

utk(xt, rt+1(s)) =rt+1(s)>xt+ (4.31)

maxxt+1,d

+t+1,d

−t+1

−∑i∈A

c(d+ti + d−ti) +

K∑j=1

Ut+2(xt+1|j)P (j|k)

s. t. 1>xt+1 +∑i∈A

c(d+ti + d−ti) = (1+ rt+1(s))>xt

xt+1,i − d+ti + d−ti = (1 + rt+1,i(s))xti, ∀i ∈ A \ 1

xt+1,d+t+1,d

−t+1 ≥ 0

In the backward step the decision at t are conditional to the state at

time t for every t and for every state. In the forward step only the current

first stage is treated differently, i.e., the state is assumed unknown. Note that,

for the forward step the recourse or future decision are also conditioned to

the states. Separating the worth functions for each state makes this approach

an approximation for the SAA problem. The entire proposed algorithm for

H2SDDP approach is presented below.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 59

Algorithm 1 H2SDDP

Require: Q0tt=2,...,T+1(Init. Upper approx.) and ε = 0.0001 (accuracy)

Initialize: i← 0, z = −∞ (Upper Bound)

while stb count < 5 and i < max do

Sample M scenarios:rt(s)1≤s≤Nt

1≤t≤T

[Foward step]

for m = 1→M do

selects scenario Sm = r0, . . . , rTfor t = 1→ T do

xmt ← argmaxxt

K∑k=1

Ut(xt|k)P (k|r[1,t]) :

1>xt = (1+ rt)>xmt−1,xt ≥ 0

end for

end for

[Backward step]

for m = 1→M , t = T → 2 do

for k = 1→ K do

for s = 1→ Nt(k) do

[utk(xmt−1, rt(s)), π

mt (rt(s))]← max

xt

rt(s)>xmt−1 +

K∑j=1

Ut+1(xt|j)P (j|k) :

1>xt = (1+ rt(s))>xmt−1,xt ≥ 0

end for

utk(xmt−1)← (1− λ) 1

Nt(k)

Nt(k)∑s=1

[utk(x

mt−1, rt(s))

]+

λ

uτtk(xt−1, rt) − 1

(1−α)Nt(k)

Nt(k)∑s=1

[utk(x

mt−1, rt(s)) −

uτtk(xt−1, rt)]−

gmt ← (1− λ) 1Nt

Nt∑s=1

γts + λ

(γt,τ + 1

(1−α)N

Nt∑s=1

ζt(rt(s))

)Ui+1tk ← xt−1 ∈ Uitk : −gmt xt−1 ≥ utk(x

mt−1)− gmt xmt−1)

end for

end for

[Upper bound update]

z ← maxx1

K∑k=1

Ui+12 (x1|k)P (k|r[T ]) : 1>x1 = (1+ r1)>x0,x1 ≥ 0

if |(z/lagr(z)− 1| < ε then

stb count ← stb count + 1

else

stb count ← 0

end if

i← i+ 1

end while

Chapter 4. Dynamic Asset Allocation Under Uncertainty 60

The dynamic asset allocation process under uncertainty embeds a se-

quence of decision at each time succeeded by the asset return realization of

the following period. In Figure 4.8, an illustrative decision tree depicts the

described process. This example represent a generic time dependence of asset

returns, since it is possible to consider different conditional probability distri-

butions of rt+1 for each given rt.

Figure 4.8: Decision tree of the generic problem with return dependence

In particular, when the return time dependence is modeled using HMM

the decision process can be represented as in Figure 4.9.

Figure 4.9: Decision tree of the problem with return dependence modeled withHMM

Note that, in Figure 4.9 the conditional allocation at a given time would

depend on the whole history of asset returns, which would lead to an intractable

optimization problem where the full tree must be represented. In Figure 4.10,

represents an computationally tractable approximation of Figure 4.9, that can

be solved by SDDP.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 61

Figure 4.10: Decision tree of the problem of our proposal

(d) Robust H2SDDP for asset allocation: Transactioncosts, temporal dependence and ambiguity aversion

As previously mentioned, it is well known that is difficult to reliably

estimate the joint probability distribution of the returns, this uncertainty

regarding the distribution can be perceive as ambiguity. Therefore, we propose

a model that is robust to such uncertainty assuming that the investor would

be averse to ambiguity.

For portfolio optimization, there are some approaches using ambiguity

aversion over mean-variance model to mitigate the returns estimation errors

[85, 86]. Our methodology is an alternative to the previously mentioned

methods in Chapter 2, to construct a robust model that reduces the sensitivity

of Markowitz’s optimal portfolio.

In the H2SDDP optimization model there is uncertainty about the

likelihood of the states, thus our approach will be to estimate the interval

of possible values for each state. It will be used those intervals on a robust

optimization to obtain a portfolio less sensitive to changes in the returns

distribution. This formulation is a generalization of the H2SDDP, presented in

the previous Section 4.3(c).

The objective function is similar to (4.28), but in this case we will use

a confidence interval for P (k|r[1,t]), that is qk ≤ P (k|r[1,t]) ≤ qk. The robust

model will choose the worst combination of pk between these values.

minq∈Q

K∑k=1

Ut(xt|k)qk (4.32)

Q =

q ∈ RK

∣∣∣∣∣ ∑k∈K

qk = 1 , q ≤ q ≤ q,

(4.33)

Chapter 4. Dynamic Asset Allocation Under Uncertainty 62

Using (4.32) and (4.33) we can define our 2-stage robust problem as

maxxt∈X(xt−1,rt)

minq

K∑k=1

Ut(xt|k)qk (4.34)

s. t.∑k

qk = 1 : z

− qk ≥ −qk ∀k ∈ 1, . . . , K : yk

qk ≥ qk ∀k ∈ 1, . . . , K : wk

x ≥ 0

The objective function has a nonlinearity due to the product of first

and second stage variables Ut(xt|k) qk. To solve the two-stage problem (4.34),

first we have to formulate the dual problem of (4.32), using the dual variables

(z, wk, yk) we have

maxz,y,w

z +K∑k=1

(qkwk − qkyk

)(4.35)

s. t. wk − yk + z ≤ Ut(xt|k) ∀k ∈ 1, . . . , K

w,y ≥ 0, z ∈ R

Adding the restrictions from the original problem and transactional costs,

we have

RPt (xt−1, r[t]) = r>t xt−1 (4.36)

maxxt,y,w,d

+t ,d−t ,z−∑i∈A

c(d+ti + d−ti) + z +

K∑k=1

(qkwk − qkyk

)s. t. 1>xt +

∑i∈A

c(d+ti + d−ti) = (1+ rt)

>xt−1

xti − d+ti + d−ti = (1 + rti)xt−1,i, ∀i ∈ A \ 1

wk − yk + z ≤ Ut(xt|k) ∀k ∈ 1, . . . , K

d+t ,d

−t ,xt,w,y ≥ 0, z ∈ R

This robust model is used only in forward procedure, since the state is

unknown, it is necessary to estimate the likelihood of the states. In backward

procedure the states are known, thus there is no uncertainty associated with

it.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 63

(e) H2SDDP for asset allocation: Transaction costs, tem-poral dependence and sell short

Sell short consist in selling an asset that is not owned by the seller,

betting on the fall in the asset price, to later buy the asset with a lower value,

thus earning the difference between the sell price and the bought price. Sell

short(selling and later purchasing) has the opposite properties of purchasing

and later selling, in which the loss is limited and gain unlimited, in short selling

the gain becomes limited but the loss is unlimited, since there is not an upper

limit for the asset price.

However, in practice it know that asset price will not rise indefinitely,

there is a reasonably limit for this value that can be evaluate, for example,

using the CVaR. Allowing sell short in asset allocation optimization models

enable the portfolio to allocate negative values for the assets, making possible

to better exploit market opportunities. In many markets the sell short involves

renting the asset until the asset is purchase and returned for who bought.

Sell short is very important for quantitative models, as with it is possible

to succeed even when the market has downward trend. For example, short

sell allows the model to obtain a positive expect value for the portfolio even

in situations when all assets have negative expected returns. Actually, using

short sell the model can take advantage of these situations, and without it the

only reasonable alternative is to invest on risk-free asset, a downward trend is

useful as upward trend when using sell short. It also allows a some leverage

behavior, by short selling an asset in order to obtain cash to invest in other

assets. With x−ti being the negative allocation of asset i, the short sell, and

x+ti the positive, also we consider the rent of sell short with penalization in

objective function costing cs

utk(xt, rt+1(s)) =r>t xt−1+ (4.37)

maxx+t ,x−t ,d

+t ,d−t

− cs∑i∈A

x−ti −∑i∈A

c(d+ti + d−ti) +

K∑j=1

Ut+2(xt+1|j)P (j|k)

s. t. 1>xt +∑i∈A

c(d+ti + d−ti) = (1+ rt)

>xt−1∑i∈A

x−ti ≤ (1+ rt)>xt−1 (4.38)

x+ti − x−ti = xti ∀i ∈ A (4.39)

xti − d+ti + d−ti = (1 + rti)xt−1,i, ∀i ∈ A \ 1

d+t ,d

−t ,x

+t ,x

−t ≥ 0

Analogous, using the constrains (4.38), (4.39) and the rent on the

Chapter 4. Dynamic Asset Allocation Under Uncertainty 64

objective this can be formulated for the forward step (4.30). Notice that the

constrain (4.38) makes this a problem without complete recourse, because the

right side can be negative. But it occurs in a very rare occasion when the whole

portfolio in negative.

4.4 Computational ExperimentsTo analyze how the proposed methods would behave in practice, we will

test and simulate with real data. The data sets used in the experiments consist

of industrial portfolios, the stocks are grouped according to the industry in

which belongs, with stocks from NYSE, AMEX and NASDAQ3. The stock

weight in the portfolio is proportional to its market value. We will use monthly

data of 5 industrial portfolios(Cnsmr, Manuf, HiTec, Hlth and Other) and daily

data for 10 industrial portfolios (NoDur, Durbl, Manuf, Enrgy, HiTec, Telcm,

Shops, Hlth, Utils and Other), and also the risk free asset with 0% of return.

Additionally, we used HMM and the k-means4 of the machine learning

library Mlpack [87] and the Latin Hypercube Sampling method from Matlab.

The multivariate Gaussian mixture of the HMM was estimated considering

the log-normal distributions of the historical returns, but in the optimization

problem was transformed to be accordingly to the real return distribution.

Algorithms were implemented in C++ language, using CPLEX5 to solve

the linear problems and also some auxiliary functions of the Armadillo library

[88]. All the experiments were conducted on Intel quad-core i5-3570 3.4 GHz

with 16GB RAM machine, only one core were used during the optimization.

The experiments are organized as follows: first it will be presented some

information about the data, used in this work, some metrics and the historical

returns of the financial time series. The following section will compare two

sampling methods, the Monte Carlo and the Latin Hypercube Sampling. Later

in this section, the model’s sensibility to variations on the parameters will be

tested, further investigating on how the results and optimal solution behaves.

Finally, out of sampling simulation is done to compare the presented methods.

(a) Data analysis

Two data sets was used on our experiments, a monthly and another daily,

both from Kenneth R. French data set 6. First tests we will use monthly data

from January 1970 to December 2014 for 5 industrial portfolios. In Table 4.1

3http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html4Used to initialize the emissions distributions of the HMM5http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/6http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html

Chapter 4. Dynamic Asset Allocation Under Uncertainty 65

we exhibit some metrics about the return series of the data set industrial

portfolios. The second data set consists in daily returns for 10 industrial

portfolio, and Table 4.2 contains a summary of the data.

Measure Cnsmr Manuf HiTec Hlth Other

Max 21.73 17.28 19.98 29.52 20.22Min -25.03 -20.80 -22.65 -20.46 -23.60Mean 1.04 1.00 0.93 1.06 0.94StdDev 4.66 4.44 5.71 4.96 5.39VaR 4.22 4.22 5.67 4.90 5.45CvaR 7.64 7.45 10.04 8.02 9.28

Table 4.1: Monthly data series for 5 industrial portfolios

Measure NoDur Durbl Manuf Enrgy HiTec Telcm Shops Hlth Utils Other

Max 10.26 9.73 10.75 19.33 16.06 14.51 10.99 11.09 14.43 11.27Min -17.03 -18.35 -20.00 -19.43 -19.98 -16.68 -16.74 -17.89 -12.86 -15.24Mean 0.05 0.04 0.05 0.05 0.04 0.05 0.05 0.05 0.04 0.04StdDev 0.91 1.37 1.09 1.36 1.48 1.14 1.10 1.12 0.88 1.19VaR 0.93 1.42 1.11 1.38 1.58 1.13 1.16 1.19 0.80 1.15CVaR 1.58 2.40 1.91 2.34 2.62 1.97 1.93 1.97 1.52 2.08

Table 4.2: Daily data series for 10 industrial portfolios

To make it easier to visualize and compare the measures, we presented

data as bar charts on Figure 4.11 and Figure 4.12.

Max Min Mean StdDev VaR CvaR

-30%

-20%

-10%

0%

10%

20%

30%

40%

Cnsmr

Manuf

HiTec

Hlth

Other

Figure 4.11: Metrics for historical monthly return series of the five industrialportfolios

Chapter 4. Dynamic Asset Allocation Under Uncertainty 66

Max Min Mean StdDev VaR CvaR

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

NoDur

Durbl

Manuf

Enrgy

HiTec

Telcm

Shops

Hlth

Utils

Other

Figure 4.12: Metrics for historical daily return series of the ten industrialportfolios

It is also important to present the financial time series of these data sets,

where Figure 4.13 and Figure 4.14 show the monthly and daily cumulative

performance from 1970 to 2014 for 5 and 10 industrial portfolio, respectively.

Date

Figure 4.13: Cumulative performance for monthly data set of the five industrialportfolios

Ret

urn

(%)

Chapter 4. Dynamic Asset Allocation Under Uncertainty 67

Date

Figure 4.14: Cumulative performance for daily data set of the ten industrialportfolios

Ret

urn

(%)

(b) Sampling Methods

As discussed before, Latin Hypercube Sampling is a promising technique

to substitute the Monte Carlo sampling in SAA techniques. Latin Hypercube

Sampling is especially important for this optimization problem, because it is

crucial to represent the distributions with least samples as possible, otherwise

it will lead to a heavy computational burden.

To compare the results of the different sampling methods, we prepared

an experiment that consists in running the optimization multiple times,

comparing the average and standard deviations of the final upper bound. It was

used the monthly data set (from 1970 to 2011), with six stages, three states,

λ 0.1, α 0.9 and no transaction cost. The test was done for Monte Carlo and

Latin Hypercube Sampling, it gradually increase number of the samples per

state 20, 100, 1000, 2500, 3000, 3500, 4000, 4500, 5000, 7000 repeating 5 times,

for each one of this values, but with different random seeds. Notice, that the

final total number of samples depends on the number of states, for example,

with 2500 samples per state and three states the problem will have a total of

7500 samples.

We did an extensive amount of tests using Monte Carlo. However,

even with 5,000 samples per stage, the standard deviation was still high. In

Figure 4.15 we compare, using the average of the upper bound for the wealth

Chapter 4. Dynamic Asset Allocation Under Uncertainty 68

and the standard deviation of it, the two sampling selection techniques for

different numbers of samples.

0 2,000 4,000 6,000

4

4.2

4.4

4.6

4.8

5

5.2·10−2

Samples

Ave

rage

UB

(wea

lth

)

MCLHS

0 2,000 4,000 6,000

0

1

2

3

4

·10−2

SamplesσUB

Figure 4.15: Comparison between Latin Hypercube Sampling (LHS) and MonteCarlo (MC)

It seems that with more samples, Monte Carlo gets a better average.

However, running with 10,000 samples it reaches 0.0407858, which is much

closer to the Latin Hypercube Sampling value with 5,000 samples, and even

with 10,000 samples the standard deviation still was very high.

As the solution of the problem is very stable using Latin Hypercube

Sampling, to get the same stability with Mote Carlo it would require much

more samples, this would increase the convergence time of the SDDP. Rep-

resenting the distribution with last realizations, as possible, can substantially

reduce computational times. Thus, we find the Latin Hypercube Sampling

more adequate sampling method to be used for this model and choose to use

2500 samples per state as it seems that using more samples does not improve

significantly the upper bound.

(c) Sensitivity Analysis

In addition to choosing the sampling method, there are many parameters

that need to be selected to be able to implement the proposed methods. We

need to select SDDP parameters such as number of stages, samples number,

number of trials, risk aversion coefficient and many others. It is also necessary

to analyze the model sensibility to variations, and if the model will behave as

expected on this tests.

The number of samples Nt can be different for each stage, but for the

sake of simplicity, the same number of sample has been considered for all

stages in this work, i.e |Nt| is the same for t ∈ H. Also, for all tests it was used

the monthly data set (from 1970 to 2011), with the default parameters as six

Chapter 4. Dynamic Asset Allocation Under Uncertainty 69

stages, three states, 2500 samples per state, λ 0.1, α 0.9 and no transaction

cost.

HMM

Moreover, there are other parameters from HMM that have to have some

initial good guessing of the states emission probabilities. We employ k-means

method to pick some centroids for every state and adopt these distributions as

the initial guess for HMM. To determine the number of states, we did a sliding

window validation for every number of states between 1 and 8, selecting the

best log-likelihood.

Figure 4.16 shows the likelihood of each state for the monthly historical

train data set.

Jan 70 Jan 73 Jan 76 Jan 79 Jan 82 Jan 85 Jan 88 Jan 91 Jan 94 Jan 97 Jan 00 Jan 03 Jan 06 Jan 090%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

3

2

1

Date

Figure 4.16: States likelihood for train data

Lik

elih

ood

We can see that the state transitions are not smooth, it seems to be

because we are using k-means as our initial emissions distributions for the

HMM, since k-means tries to minimize the number of intersections in the

clusters. We should expect more diversified results if the samples were classify

instead of using clusters.

Impact of the Risk Aversion

The risk aversion coefficient (λ) also has big impact in the upper bound,

that is the certainty equivalent of the portfolio, which can be interpreted as the

fund worth. In Figure 4.17 we present the upper bound for different λ values,

for monthly returns on left and on right for daily returns. This results shows

how the value of the portfolio changes for investors with different aversion to

risk.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 70

0 0.1 0.2 0.3 0.4

0

2

4

6

·10−2

λ

Up

per

Bou

nd

(Wea

lth

)

0.02 0.04 0.06 0.08 0.1

0

1

2

3

·10−3

λ

Up

per

Bou

nd

(Wea

lth

)

Figure 4.17: Impact of the risk aversion coefficient (λ) on the upper bound formonthly returns and daily returns, respectively on the left and right

As expected, to have the same return as in monthly data set, the risk

aversion coefficient has to be lower for the daily returns. For daily returns when

λ = 0.1 the optimal solution is to allocate everything on the risk-free asset,

the correspondent value for monthly returns is 0.3. The risk aversion coefficient

should get higher when allowing short sell.

Convergence and trials

Since the number of possible scenarios increases exponentially with the

number of stages, it is important to investigate how the number of stages affects

the computational time, convergence and also the final solution. As the number

of stages increase the computational time should also rise, since H2SDDP needs

more iterations to converge. In Table 4.3 we present upper bound, time and

number of cuts according to number of states, time is presented in minutes.

The computational time needed to converge the SDDP get really high. Observe

n. stages upper bound time n. cuts

12 0.0772493 41 5024 0.15515 149 7548 0.324179 356 8296 0.754923 1088 109

Table 4.3: Number of stages and the computational time

that with twice more stages the number of cuts increases slightly and the

computational time increases substantially.

We did some prior tests to check whether it would be better to use many

trials or only one trial at a time. This also determine the number of cuts added

at each stage. Three ways of choosing the number of trials were tested; first

Chapter 4. Dynamic Asset Allocation Under Uncertainty 71

using only one trial, second using 5 trials and, finally, using 5 trials as the upper

bound becomes stable for 10 iterations. From our limit experiments, using more

than one trial, i.e., adding multiple cuts, increase the convergence time even

with the alternative approach using dynamic number of trials. It seems better

to do C iterations adding one cut per iteration than to add C cuts on one

iteration, for the convergence of the algorithm. The real advantage of adding

multiple cuts could be on the quality of the worth function approximation,

this could enhance stability of primal solutions.

The used method to test the convergence of SDDP was the stabilization

of the upper bound. Whether, the upper bound of 10 iterations ago zi−10

is stable for 5 iterations, i.e., 15 iterations without change greater than ε.

Notice that the upper bound of the iteration i − 10 is compared with the

current i upper bound value. We opted for a tighter convergence criteria with

ε = 0.0001 than do more tries. To illustrate the convergence of the H2SDDP,

Figure 4.18 exhibits the upper bound changes, in percentage, for optimization

of the H2SDDP with 96 periods.

Figure 4.18: Convergence of the Upper Bound for monthly data set

Transactional costs

Another interesting subject is the impact of the transactional costs. First,

we present how of the upper bound is sensible to variations of the transactional

costs on Figure 4.19. Later we will discuss further implications of taking the

transactional costs into account. It was used the monthly data set (from 1970

Chapter 4. Dynamic Asset Allocation Under Uncertainty 72

to 2011), with six stages, three states, 2500 samples per state, λ 0.1, α 0.9 and

no transaction cost.

0 0.5 1 1.5 2 2.5 32.5

3

3.5

4

·10−2

Transactional cost(%)

Upp

erB

ound

(Wea

lth)

Figure 4.19: Impact of the transactional costs on the upper bound

It is evidently that the transactional costs can really impact the optimal

solution. Since the return is uncertain but the transactional cost is not,

higher transactional cost makes not worth to do some small changes on the

investments, thus the model only invest when the expect return is higher

enough to mitigate the costs involved.

Further analyzing how the transactional costs will affect the optimal solu-

tion. The portfolio allocation can be seen on Figure 4.20 with no transaction

costs, and in Figure 4.21 with transactional cost equal to 0.1 %. Even at a

cost of 0.1 %, smaller deviations are avoided when the model does not expect

a significant gain with changing the portfolio. In Figure 4.20 and Figure 4.21

we can clearly see the portfolio stabilization effect caused by transaction costs,

it is evident the resemblance to a regularization L1.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 73

Jan 12 Apr 12 Jul 12 Oct 12 Jan 13 Apr 13 Jul 13 Oct 13 Jan 14 Apr 14 Jul 14 Oct 140%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Other

Hlth

HiTec

Manuf

Cnsmr

Risk Free

Date

Figure 4.20: Portfolio allocation without transactional costs

Por

tfoli

oP

erce

nta

ge

Jan 12 Apr 12 Jul 12 Oct 12 Jan 13 Apr 13 Jul 13 Oct 13 Jan 14 Apr 14 Jul 14 Oct 140%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Other

Hlth

HiTec

Manuf

Cnsmr

Risk Free

Date

Figure 4.21: Portfolio allocation with transactional costs

Por

tfol

ioP

erce

nta

ge

(d) Models Evaluation

This section present the final results of simulations on out of sample

data, in order to do evaluate the performance of the models. We will compare

H2SDDP with different approaches and present the results for ambiguity

aversion model. The parameters for the experiments on this sections are: 2500

Chapter 4. Dynamic Asset Allocation Under Uncertainty 74

samples per stage, three states form the HMM, lambda 0.1, alpha 0.9 and

transactional cost 0.1%, it will be mentioned if there is any changes on this

parameters. As the initial wealth is one, this value can also be interpreted as

the return of the policy in percentage.

Experiment with Monthly Data Set 2012 to 2014

On the first tests we are going to use the monthly data set, training with

dates from 1970 to 2011 and simulating with 2012 to 2014 (36 months). In

H2SDDP simulations the train data set(1970 to 2011) was used to evaluate

the policy, then the policy was used to evaluate the performance on test data

set(2012 to 2014).

To compare the performance of the policy accordingly to the risk aversion

coefficient, we alternate the λ value to evaluate what would be the behavior

in out of samples data, the result is presented in Table 4.4.

λ CVaR Ret.0.2 4.16 155.720.1 4.14 158.340.09 4.18 160.790.08 4.04 162.290.07 4.06 166.350.06 4.12 165.040.05 4.31 170.450.025 4.75 171.36

Table 4.4: CVaR and return values of the simulated polices for different riskcoefficients, results in percentage

In order to analyze and compare the proposed approaches, besides of

simulating the previous presented models we also evaluate the results for the

equally weight portfolio and the myopic strategy. One can easily underestimate

the value of the equally weighted portfolio but in practical situations they

outperform most of the traditional optimization approaches [89]. Important to

mention, since we are considering equal distribution of returns for each stage,

the myopic method has the same portfolio in percentage for the entire period,

then the myopic portfolio was establish with a 2-stage optimization.

Figure 4.22 show the result of the simulations. This is an experiment

using the monthly data set comparing H2SDDP with λ 0.1 and 0.05 with the

myopic and equally weight policies.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 75

Date

Figure 4.22: Comparing the different methods for asset allocation data form2012 to 2014

Wea

lth

Further investigation is required to interpret the results of Figure 4.22.

Inspecting the assets cumulative returns on Figure 4.23 for the same time

period we can observe that this was an interval of strong upward trend and,

furthermore, all assets has also very similar behavior for this period. This

hypothesis also is sustained by results presented on Table 4.4, where the

greatest return is obtained with λ = 0.025. In this scenario, it is expected

to have similar results for all the presented methods.

Date

Figure 4.23: Cumulative returns series from January 2012 to December 2014

Wea

lth

Chapter 4. Dynamic Asset Allocation Under Uncertainty 76

Experiment with Monthly Data 2007 to 2014

To test this interpretation we develop a new test case now simulating on

data from 2007 to 2014, as in this period there is more volatility, and changes in

trends, the dynamic policy should be capable of changing the portfolio to adapt

to this variations. The models uses samples from 1970 to 2006 and 96 stages,

after finding the optimal solution the approximations of the expected worth

functions are used on the test with 96 months, January 2007 to December

2014. The cumulative performance for the industrial portfolio from this period

is exhibit in Figure 4.24. The transactional costs was account to all the models,

for those that do not consider in the optimization model it was discounted the

costs after the optimization.

Date

Wea

lth

Figure 4.24: Returns series from January 2007 to December 2014

In this experiment, on Figure 4.25 beside the H2SDDP, Equally weighted

and Myopic models, it is also presented the performance of the Robust model,

using and interval of 10%, i.e. 0.9 × P (k|r[1,t]) ≤ qk ≤ 1.1 × P (k|r[1,t]), ∀k ∈1, . . . , K.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 77

Date

Figure 4.25: Comparing the different methods for asset allocation data form2007 to 2014

Wea

lth

The H2SDDP has almost 20% and 50% more income than the Myopic

and equal policies, respectively. Analyzing more carefully the Figure 4.25, we

can see that as soon as the H2SDDP observe the downward trend it allocates

everything on the risk-free asset. However, there is a delay to see the upward

trend of the market.

Figure 4.26 compares the state of the market with the performance of

the H2SDDP. With it, becomes clear that the model can react to changes in

the market behavior, even in a downtrend, this show how important is to have

a constant evaluation of the portfolio. Considering the choices on Figure 4.25

the best investment for this period is H2SDDP. In this scenario Robust model

perform very similar to the H2SDDP, to estimate property the interval require

further investigation.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 78

Date

Figure 4.26: Comparing the market states with the performance of theH2SDDP for data form 2007 to 2014

Wea

lth

Moreover, to better judge the performance of the models over different

periods, we simulate the investment for 6 consecutively months in each strategy

as if it was a fund. Figure 4.27 presents the semiannual cumulative performance

for every month.

Date

Figure 4.27: Trailing returns for asset allocation methods with monthly dataform 2007 to 2014

Wea

lth

Using short sell the model can take advantage of these situations, and

without it the only reasonable action is to invest on risk-free asset. Therefore,

Chapter 4. Dynamic Asset Allocation Under Uncertainty 79

we want to verify if the use of short selling can help the model improve the

policy return. In order to test how H2SDDP will perform with short sell, we

implement a model that allow short selling but limit it on the total wealth of

the portfolio in the last period. On Figure 4.28 we compare the performance

of H2SDDP, allowing short selling, with other methods for the same period.

Date

Figure 4.28: Comparing the performance of H2SDDP with sell short sell shortwith other methods, monthly data form 2007 to 2014

Wea

lth

The sell short model has by far the highest cumulative return of the

models and also perform similar to the H2SDDP, without short sell, in the

downward trend, investing most of the portfolio on the risk free asset. However,

the sell short took longer to identify the downward trend it also react very fast

when the trend changed.

It seems that allowing short sell makes the model enhance the high

returns, short selling the more stable assets to invest more on asset with high

expected returns. The final cumulative return with sell short(527%) was more

than twice the value of the second best model H2SDDP (218%).

In the trailing returns on Figure 4.29 we can observe that Sell Short

H2SDDP has the higher return for most of the period, but this is also a more

risk model than the H2SDDP without short selling. Although, the model has

it lowest’ possible return at the end of the year 2011 the Myopic and Equal

models reaches lower returns than this, in the begin of 2009. Is is also possible

to decrease the λ, that was 0.1, to obtain more stable performance.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 80

Date

Figure 4.29: Trailing returns of H2SDDP with sell short sell and other methods,monthly data form 2007 to 2014

Wea

lth

Experiment with Daily Data Set

To be able to observe how the model will behavior with more assets and

more volatility, we perform an experiment with daily returns from 10 industrial

portfolios. It was used data from 2007 to 2014, it not possible to create cuts

for 2012 days, thus the daily data set was divide in intervals of 22 days, so the

optimization has 22 cuts and every 22 day the model will be retrained. The

only difference in the parameters was the λ = 0.03.

Figure 4.30 shows the results for the H2SSP and, Equal weight and

Myopic, the figure emphasizes the importance of a dynamic allocation, in

which the model can change allocation when facing critical events, without

it the portfolio depends on the tendency of the market to get good results. By

the end of 2008 the cumulative return for the period was 113.16%, 71.16% and

73.98%, for the H2SSP, Equal weight and Myopic, respectively. Figure 4.31 is

trailing returns for this experiment.

Chapter 4. Dynamic Asset Allocation Under Uncertainty 81

Date

Figure 4.30: Cumulative performance for asset allocation methods with dailydata set form 2007 to 2014

Wea

lth

Date

Figure 4.31: Trailing returns for asset allocation methods with daily data setform 2007 to 2014

Wea

lth

Although, H2SDDP strategy fails to predict the upward tendency after

2008, it has clearly a more controlled risk than the other strategies; it has

lower CVaR and standard deviation for this period (2008 to 2014). This can

be evidence that the model should be retrained more frequently than 22 days.

Besides that, some further investigation is necessary to better compare these

strategies using different data sets with more diversity.

5Conclusions and Future Works

This thesis aims at contributing for the optimization of models for asset

allocation under uncertainty. Different techniques are proposed herein to assist

in the asset allocation problem. Experiments were performed to evaluate the

behavior of ours approaches and they turn down to be very promising for both

of the presented cases.

It is a challenge to find the portfolio that best suit the investor’s needs.

The uncertainty surrounding future returns and the behavior of the assets is a

delicate issue in its own, but when the irrationality caused by human emotions

is added to the equation, the problem becomes even more difficult.

In the traditional asset allocation model, the investor has to choose the

return or the risk of the portfolio. Such mean-variance model has several issues,

however and, as a consequence, we have some proposals for improving the

mean-variance model by making it less sensitive to such issues. One of these

proposals is the Black-Litterman, which combines the investor’s vision with

the mean-variance model and the CAPM theory.

Although it is undeniable that the Black-Litterman technique is able

to mitigate most of the shortcomings of the traditional method proposed by

Markowitz, the construction of views can be a confusing process and depends

largely on the investor’s ability to quantify something extremely subjective.

We propose a new way to solve this problem by using questionnaires

and the ZAPROS-III method to construct the views. Techniques like the VDA

enable us to transform the answers provided in the sets of questions into views

of the Black-Litterman model. A case study based on Brazilian stocks was

conducted as a demonstration of how to use the methodology, and the results

were as expected.

The most important advantage of the proposed solution is that it allows

an investor to create his own portfolio, based on his own opinion, and without

the help of an expert. The final allocation will be optimized considering the

investor vision about the future, this typically will create a more diversified

and less risk portfolio. It also makes it easier for the investor to manifest his

own opinion and facilitates the process, allowing the investor to modify his

portfolio more frequently.

Chapter 5. Conclusions and Future Works 83

While we believe that this approach presents a lot of advantages, it also

has its limitations. The ZAPROS method is not adequate for a larger number of

alternatives or criteria, rendering this method applicable only to cases involving

a few stocks and questions. This shortcoming is a relative one, as the focus of

this proposal has always been to create a simple method, we believe it is not

reasonable to ask an investor too many questions. Besides, even though only

two views were constructed in the scope of this work, a larger number of views

may be constructed in order to represent other visions.

It would be interesting to have further work done in the sense of proposing

or developing new ways to improve the acquisition of the necessary parameters

for the Black-Litterman views, like the expected return. Future work could also

put other risk functions to the test and analyze the model’s behavior. Also the

construction of the views can be done in multilevel using ORCLASS, or some

other similar method, to select the assets of each view and then use ZAPROS

to determine the weight of the assets in the views.

Aiming at a multistage solution that also analyzes the assets in a more

qualitative way, this work proposes the use of stochastic programming for assets

allocation. Most of the proposed solutions in the literature cannot handle a

large number of stages, but the SDDP method has been efficiently used for

large-scale problems for several years now, and was thus chosen in this work

for the multistage asset allocation.

Largely used for hydrothermal scheduling problems, SDDP is specially

suited for large-scale problems. In it, the algorithm implicitly represent the

scenarios tree, building at each iteration better approximations for the worth

function for each stage. The time independence hypothesis of the original

proposed SDDP makes it non-applicable to some types of problems. While

some proposals to adapt the SDDP to consider the time dependence already

exist, none of these can be directly applied to the asset allocation problem.

Hence the methodology in this work, which models this dependency by using

hidden Markovian states. In addition, as there is some ambiguity in the joint

probability of the returns’ distribution, a more general method with ambiguity

aversion has also been developed.

In the experiments section we present the metrics for the data sets, com-

parison between Monte Carlo and Latin Hypercube Sampling, the sensitivity

analyses of the model, showing how H2SDDP performs with different para-

meters, and the final simulations of the proposed methods with out of sample

data. Besides showing how the optimization models would perform as an actual

fund, the experiments emphasizes the importance continuous re-evaluation of

the portfolio.

Chapter 5. Conclusions and Future Works 84

As the use of multistage stochastic techniques for portfolio optimization

becomes more and more widespread, both fields experience constant changes

and new ideas and methods are often proposed. Therefore, several ideas for

possible future work arose during the development of this thesis. One of them

would be to constraint the CVaR, aiming at maximizing the expected value

of the portfolio. The advantage in this case would be to have an easy way

to calculate the lower bound, as well as an easier way of interpreting the

model’s characteristics, risk constraint and objective function. Also, choosing

a maximum value for the CVaR makes interpretation easier and is a more

natural choice than the risk aversion index.

The multistage CVaR constraint has been suggested for the hydrothermal

scheduling problem [90], but adding this constraint to such model my render

it unfeasible. Thus, the normal approach for such cases is to use Lagrangian

relaxation to penalize violations, of the constraint, in the objective function.

In asset allocation problems, however, such feasibility issue is not expected

because allocating everything in the risk free asset is always feasible (just has to

have γ bigger than the transactional cost). This makes the model significantly

more straightforward and not imposing any changes to the objective function.

The main difference in this model is that the optimization needs to be

done in two-stages. The variables yti must be created, representing a vector

with the amount invested in each asset in t. Adding these ideas with H2SDDP,

we acquire

Qt(yt|k) =

maxxt,d

+t ,d−t ,δ,z− c

A∑i=1

(d+ti + d−ti

)+

K∑j=1

P (j|k)Nt∑s=1

P (s|j)

r>t+1(s)xt +Qt+1((1 + rt+1(s))>xt|j)

s. t.A∑i=1

xti + cA∑i=1

(d+ti + d−ti

)=

A∑i=1

yti

xti − d+ti + d−ti = yti ∀ i ∈ A \ 1

z +

K∑j=1

Nt∑s=1

P (j|k)P (s|j)δjs

(1−α)+ c

A∑i=1

(d+ti + d−ti

)≤ γ

(A∑i=1

yti

)δjs ≥ −z − r>t+1(s)xt ∀j ∈ 1, . . . , K, s ∈ 1, . . . , Ntδ,xt,d

+t ,d

−t ≥ 0

where P (s|j) = 1Nt

. Further experimentation will be made with this model in

near future, in order to compare it with the other models developed in this

thesis.

The estimation of the joint distributions of the returns is especially

Chapter 5. Conclusions and Future Works 85

problematic in asset allocation problems, but there are some approaches that

may be used in order to minimize this issue. One of them is to incorporate

robust approaches to the models proposed here by using the Bertsimas ideas

on the probability states of the Markov model. Such proposal is an extension

of the ambiguity aversion model presented in Section 4.3(d).

In addition, it must be said that some of the concepts proposed here

could not be computed due to the long computational time needed to run the

experiments, which is why it would be interesting to have some proposal to

reduce the algorithms’ execution time. In this regard, our research indicates

that a first and straightforward possible solution would be to design parallel

algorithms, making better use of modern computers that currently have

multiple cores. Another alternative would be to create heuristics to remove

some of the cutting planes’ cuts, a solution that has already shown some

improvements in other researches.

There is also the need to better understand the behavior of the HMM in

practice, and how to better estimate the initial probabilities. The method k-

means is a suitable method for this situation, but may have unwanted collateral

effects. We could not find a better alternative in literature, but it is possible

to test with manual classification.

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AAppendix

A.1 QuestionnairesThe questions that were mention on section 3.4 are presented in Table A.1

and Table A.2. This first survey on Table A.1 is specific about companies and

it is used to construct the assets view. Six questions about the companies

covering issues such as risk, stability, innovation and profitability.

Criteria Possible valuesA. Risk A1. Low risk

A2. Medium riskA3. High risk

B. Stability B1. Company with years of market experienceand traditionB2. Company with some market timeB3. Company with little market time

C. Expected growth C1. Company with promising future and ac-celerated growthC2. Company that is expected some growthC3. Company which is not expected growth

D. Innovation D1. The company invests heavily in R&D andalways comes up with new ideasD2. The company invests little inR&D and new ideas usually ariseD3. The company does not invest in R&D andalmost never comes up new ideas

E. Profitability E1. Company always transfers profits toshareholdersE2. Company usually transfers profits toshareholdersE3. Company almost never transfers profits toshareholders

F. Employees F1. Highly qualified employees that are alwaysmotivatedF2. Good employees that are usually motiv-atedF3. Employees without much qualification andlack of motivation

Table A.1: Questionnaire about the stocks

Appendix A. Appendix 96

Second survey on sectors Table A.2 is more simple and relative to the

sectors of the market.

Criteria Possible values

A. Internal market A1. Very favorable to industry

A2. Favorable to industry

A3. Not favorable to industry

B. External market B1. Very favorable to industry

B2. Favorable to industry

B3. Not favorable to industry

C. Expected growth C1. It is expected a high growth

C2. It is expected some growth

C3. It is expected little or none growth

Table A.2: Questionnaire about the sectors

A.2 Myopic proveProposition 1 For the portfolio selection with no transaction cost and without

temporal dependence the myopic policy is optimal, i.e. Qt(·) is positive homo-

geneous. As a result, in this case one only needs to consider the return of the

next moment to decide on an investment.

Proof : To prove by induction we will first show for the base case that QT−1(·),then assuming that the proposition holds up to Qt+1(·) and deduce Qt(·)

For the base case the optimal value on T depends on WT−1 and r[T−1],it

will be used RT = 1+ rT , thus we have

QT−1(WT−1, r[T−1]) = maxxt−1

ψT−1[R>T xt−1|r[T−1]]

s. t. 1>xt−1 = WT−1

xt−1 ≥ 0

For t = T − 2, . . . , 0

Qt(Wt, r[t]) = maxxt

ψt[Qt+1(Wt, r[t])]

s. t. 1>xt = Wt

rt+1(r[t])>xt = Wt+1

xt ≥ 0

Appendix A. Appendix 97

As RT is independent from the past r[t−1]

QT−1(WT−1) = maxxt−1

ψ[R>T xt−1]

s. t. 1>xt−1 = WT−1

xt−1 ≥ 0

For t = T − 2, . . . , 0

Qt(Wt) = maxxt

ψ[Qt+1(Wt+1)]

s. t. 1>xt = Wt

R>t+1xt = Wt+1

xt ≥ 0

Let yt = xt

Wt, ∀t = 0, . . . , T − 1

QT−1(WT−1) = maxyT−1

ψ[R>T yT−1WT−1]

s. t. 1>yT−1 = 1

yT−1 ≥ 0

QT−1(WT−1) = WT−1 ×maxyT−1

ψ[R>T yT−1]

s. t. 1>yT−1 = 1

yT−1 ≥ 0

QT−1(WT−1) = WT−1QT−1(1)

Inductive hypothesis, assuming that this proposition is valid for t+ 1

Qt+1(Wt+1) = Wt+1 ×T−1∏t′=t+1

Qt′(1)

Inductive step, for t

Qt(Wt) = maxxt

ψ[Qt+1(Wt+1)]

s. t. 1>xt = Wt

r>t+1xt = Wt+1

xt ≥ 0

Appendix A. Appendix 98

by our inductive hypothesis

ψ[Qt+1(Wt+1)] =ψ[Wt+1 ×T−1∏t′=t+1

Qt′(1)] = ψ[r>t+1xt ×T−1∏t′=t+1

Qt′(1)]

Qt(Wt) =T−1∏t′=t+1

Qt′(1)×maxxt

ψ[r>t+1xt]

s. t. 1>xt = Wt

xt ≥ 0 (A.1)

Swapping xt for yt ×Wt

Qt(Wt) =T−1∏t′=t+1

Qt′(1)×maxyt

ψ[r>t+1yt ×Wt]

s. t. 1>yt = 1

yt ≥ 0

Qt(Wt) = Wt ×T−1∏t′=t+1

Qt′(1)×maxyt

ψ[r>t+1yt]

s. t. 1>yt = 1

yt ≥ 0

Qt(Wt) = Wt ×Qt(1)×T−1∏t′=t+1

Qt′(1)

Qt(Wt) = Wt ×T−1∏t′=t

Qt′(1)

Note the first stage problem t = 1 using formulation (A.1)

Q0(W0) =T−1∏t′=2

Qt′(1)×maxx1

ψ[R>2 x0]

s. t. 1>x0 = W0

x1 ≥ 0

As the first stage decisions x0 are independent from Qt(1) ∀t ∈ 2, . . . , T − 1,to retrieve optimal solution it is only necessary to solve the problem below

ignoring future returns.

maxx1

ψ[R>2 x1] (A.2)

Appendix A. Appendix 99

s. t. 1>x1 = W1

x0 ≥ 0

Likewise, it is possible to obtain optimal solution for Qt(Wt) ∀t ∈ T .